CRYPTOGRAPHY AND
NETWORK SECURITY
PRINCIPLES AND PRACTICE
SEVENTH EDITION
GLOBAL EDITION

William Stallings

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3

CONTENTS
Notation 10

Preface 12

About the Author 18

PART ONE: BACKGROUND 19

Chapter 1 Computer and Network Security Concepts 19

1.1 Computer Security Concepts 21
1.2 The OSI Security Architecture 26
1.3 Security Attacks 27
1.4 Security Services 29
1.5 Security Mechanisms 32
1.6 Fundamental Security Design Principles 34
1.7 Attack Surfaces and Attack Trees 37
1.8 A Model for Network Security 41
1.9 Standards 43
1.10 Key Terms, Review Questions, and Problems 44

Chapter 2 Introduction to Number Theory 46

2.1 Divisibility and the Division Algorithm 47
2.2 The Euclidean Algorithm 49
2.3 Modular Arithmetic 53
2.4 Prime Numbers 61
2.5 Fermat’s and Euler’s Theorems 64
2.6 Testing for Primality 68
2.7 The Chinese Remainder Theorem 71
2.8 Discrete Logarithms 73
2.9 Key Terms, Review Questions, and Problems 78
Appendix 2A The Meaning of Mod 82

PART TWO: SYMMETRIC CIPHERS 85

Chapter 3 Classical Encryption Techniques 85

3.1 Symmetric Cipher Model 86
3.2 Substitution Techniques 92
3.3 Transposition Techniques 107
3.4 Rotor Machines 108
3.5 Steganography 110
3.6 Key Terms, Review Questions, and Problems 112

Chapter 4 Block Ciphers and the Data Encryption Standard 118

4.1 Traditional Block Cipher Structure 119
4.2 The Data Encryption Standard 129
4.3 A DES Example 131
4.4 The Strength of DES 134

4 CONTENTS

4.5 Block Cipher Design Principles 135
4.6 Key Terms, Review Questions, and Problems 137

Chapter 5 Finite Fields 141

5.1 Groups 143
5.2 Rings 145
5.3 Fields 146
5.4 Finite Fields of the Form GF(p) 147
5.5 Polynomial Arithmetic 151
5.6 Finite Fields of the Form GF(2n) 157
5.7 Key Terms, Review Questions, and Problems 169

Chapter 6 Advanced Encryption Standard 171

6.1 Finite Field Arithmetic 172
6.2 AES Structure 174
6.3 AES Transformation Functions 179
6.4 AES Key Expansion 190
6.5 An AES Example 193
6.6 AES Implementation 197
6.7 Key Terms, Review Questions, and Problems 202
Appendix 6A Polynomials with Coefficients in GF(28) 203

Chapter 7 Block Cipher Operation 207

7.1 Multiple Encryption and Triple DES 208
7.2 Electronic Codebook 213
7.3 Cipher Block Chaining Mode 216
7.4 Cipher Feedback Mode 218
7.5 Output Feedback Mode 220
7.6 Counter Mode 222
7.7 XTS-AES Mode for Block-Oriented Storage Devices 224
7.8 Format-Preserving Encryption 231
7.9 Key Terms, Review Questions, and Problems 245

Chapter 8 Random Bit Generation and Stream Ciphers 250

8.1 Principles of Pseudorandom Number Generation 252
8.2 Pseudorandom Number Generators 258
8.3 Pseudorandom Number Generation Using a Block Cipher 261
8.4 Stream Ciphers 267
8.5 RC4 269
8.6 True Random Number Generators 271
8.7 Key Terms, Review Questions, and Problems 280

PART THREE: ASYMMETRIC CIPHERS 283

Chapter 9 Public-Key Cryptography and RSA 283

9.1 Principles of Public-Key Cryptosystems 285
9.2 The RSA Algorithm 294
9.3 Key Terms, Review Questions, and Problems 308

CONTENTS 5

Chapter 10 Other Public-Key Cryptosystems 313

10.1 Diffie-Hellman Key Exchange 314
10.2 Elgamal Cryptographic System 318
10.3 Elliptic Curve Arithmetic 321
10.4 Elliptic Curve Cryptography 330
10.5 Pseudorandom Number Generation Based on an Asymmetric Cipher 334
10.6 Key Terms, Review Questions, and Problems 336

PART FOUR: CRYPTOGRAPHIC DATA INTEGRITY ALGORITHMS 339

Chapter 11 Cryptographic Hash Functions 339

11.1 Applications of Cryptographic Hash Functions 341
11.2 Two Simple Hash Functions 346
11.3 Requirements and Security 348
11.4 Hash Functions Based on Cipher Block Chaining 354
11.5 Secure Hash Algorithm (SHA) 355
11.6 SHA-3 365
11.7 Key Terms, Review Questions, and Problems 377

Chapter 12 Message Authentication Codes 381

12.1 Message Authentication Requirements 382
12.2 Message Authentication Functions 383
12.3 Requirements for Message Authentication Codes 391
12.4 Security of MACs 393
12.5 MACs Based on Hash Functions: HMAC 394
12.6 MACs Based on Block Ciphers: DAA and CMAC 399
12.7 Authenticated Encryption: CCM and GCM 402
12.8 Key Wrapping 408
12.9 Pseudorandom Number Generation Using Hash Functions and MACs 413
12.10 Key Terms, Review Questions, and Problems 416

Chapter 13 Digital Signatures 419

13.1 Digital Signatures 421
13.2 Elgamal Digital Signature Scheme 424
13.3 Schnorr Digital Signature Scheme 425
13.4 NIST Digital Signature Algorithm 426
13.5 Elliptic Curve Digital Signature Algorithm 430
13.6 RSA-PSS Digital Signature Algorithm 433
13.7 Key Terms, Review Questions, and Problems 438

PART FIVE: MUTUAL TRUST 441

Chapter 14 Key Management and Distribution 441

14.1 Symmetric Key Distribution Using Symmetric Encryption 442
14.2 Symmetric Key Distribution Using Asymmetric Encryption 451
14.3 Distribution of Public Keys 454
14.4 X.509 Certificates 459

6 CONTENTS

14.5 Public-Key Infrastructure 467
14.6 Key Terms, Review Questions, and Problems 469

Chapter 15 User Authentication 473

15.1 Remote User-Authentication Principles 474
15.2 Remote User-Authentication Using Symmetric Encryption 478
15.3 Kerberos 482
15.4 Remote User-Authentication Using Asymmetric Encryption 500
15.5 Federated Identity Management 502
15.6 Personal Identity Verification 508
15.7 Key Terms, Review Questions, and Problems 515

PART SIX: NETWORK AND INTERNET SECURITY 519

Chapter 16 Network Access Control and Cloud Security 519

16.1 Network Access Control 520
16.2 Extensible Authentication Protocol 523
16.3 IEEE 802.1X Port-Based Network Access Control 527
16.4 Cloud Computing 529
16.5 Cloud Security Risks and Countermeasures 535
16.6 Data Protection in the Cloud 537
16.7 Cloud Security as a Service 541
16.8 Addressing Cloud Computing Security Concerns 544
16.9 Key Terms, Review Questions, and Problems 545

Chapter 17 Transport-Level Security 546

17.1 Web Security Considerations 547
17.2 Transport Layer Security 549
17.3 HTTPS 566
17.4 Secure Shell (SSH) 567
17.5 Key Terms, Review Questions, and Problems 579

Chapter 18 Wireless Network Security 581

18.1 Wireless Security 582
18.2 Mobile Device Security 585
18.3 IEEE 802.11 Wireless LAN Overview 589
18.4 IEEE 802.11i Wireless LAN Security 595
18.5 Key Terms, Review Questions, and Problems 610

Chapter 19 Electronic Mail Security 612

19.1 Internet Mail Architecture 613
19.2 Email Formats 617
19.3 Email Threats and Comprehensive Email Security 625
19.4 S/MIME 627
19.5 Pretty Good Privacy 638
19.6 DNSSEC 639
19.7 DNS-Based Authentication of Named Entities 643
19.8 Sender Policy Framework 645
19.9 DomainKeys Identified Mail 648

CONTENTS 7

19.10 Domain-Based Message Authentication, Reporting, and Conformance 654
19.11 Key Terms, Review Questions, and Problems 659

Chapter 20 IP Security 661

20.1 IP Security Overview 662
20.2 IP Security Policy 668
20.3 Encapsulating Security Payload 673
20.4 Combining Security Associations 681
20.5 Internet Key Exchange 684
20.6 Cryptographic Suites 692
20.7 Key Terms, Review Questions, and Problems 694

APPENDICES 696

Appendix A Projects for Teaching Cryptography and Network Security 696

A.1 Sage Computer Algebra Projects 697
A.2 Hacking Project 698
A.3 Block Cipher Projects 699
A.4 Laboratory Exercises 699
A.5 Research Projects 699
A.6 Programming Projects 700
A.7 Practical Security Assessments 700
A.8 Firewall Projects 701
A.9 Case Studies 701
A.10 Writing Assignments 701
A.11 Reading/Report Assignments 702
A.12 Discussion Topics 702

Appendix B Sage Examples 703

B.1 Linear Algebra and Matrix Functionality 704
B.2 Chapter 2: Number Theory 705
B.3 Chapter 3: Classical Encryption 710
B.4 Chapter 4: Block Ciphers and the Data Encryption Standard 713
B.5 Chapter 5: Basic Concepts in Number Theory and Finite Fields 717
B.6 Chapter 6: Advanced Encryption Standard 724
B.7 Chapter 8: Pseudorandom Number Generation and Stream Ciphers 729
B.8 Chapter 9: Public-Key Cryptography and RSA 731
B.9 Chapter 10: Other Public-Key Cryptosystems 734
B.10 Chapter 11: Cryptographic Hash Functions 739
B.11 Chapter 13: Digital Signatures 741

References 744

Credits 753

Index 754

8 CONTENTS

ONLINE CHAPTERS AND APPENDICES1

PART SEVEN: SYSTEM SECURITY

Chapter 21 Malicious Software

21.1 Types of Malicious Software (Malware)
21.2 Advanced Persistent Threat
21.3 Propagation—Infected Content—Viruses
21.4 Propagation—Vulnerability Exploit—Worms
21.5 Propagation—Social Engineering—Spam E-mail, Trojans
21.6 Payload—System Corruption
21.7 Payload—Attack Agent—Zombie, Bots
21.8 Payload—Information Theft—Keyloggers, Phishing, Spyware
21.9 Payload—Stealthing—Backdoors, Rootkits
21.10 Countermeasures
21.11 Distributed Denial of Service Attacks
21.12 References
21.13 Key Terms, Review Questions, and Problems

Chapter 22 Intruders

22.1 Intruders
22.2 Intrusion Detection
22.3 Password Management
22.4 References
22.5 Key Terms, Review Questions, and Problems

Chapter 23 Firewalls

23.1 The Need for Firewalls
23.2 Firewall Characteristics and Access Policy
23.3 Types of Firewalls
23.4 Firewall Basing
23.5 Firewall Location and Configurations
23.6 References
23.7 Key Terms, Review Questions, and Problems

PART EIGHT: LEGAL AND ETHICAL ISSUES

Chapter 24 Legal and Ethical Aspects

24.1 Cybercrime and Computer Crime
24.2 Intellectual Property
24.3 Privacy
24.4 Ethical Issues
24.5 Recommended Reading
24.6 References
24.7 Key Terms, Review Questions, and Problems
24.A Information Privacy

1Online chapters, appendices, and other documents are at the Companion Website, available via the
access card at the front of this book.

CONTENTS 9

Appendix C Sage Exercises

Appendix D Standards and Standard-Setting Organizations

Appendix E Basic Concepts from Linear Algebra

Appendix F Measures of Secrecy and Security

Appendix G Simplified DES

Appendix H Evaluation Criteria for AES

Appendix I Simplified AES

Appendix J The Knapsack Algorithm

Appendix K Proof of the Digital Signature Algorithm

Appendix L TCP/IP and OSI

Appendix M Java Cryptographic APIs

Appendix N MD5 Hash Function

Appendix O Data Compression Using ZIP

Appendix P PGP

Appendix Q The International Reference Alphabet

Appendix R Proof of the RSA Algorithm

Appendix S Data Encryption Standard

Appendix T Kerberos Encryption Techniques

Appendix U Mathematical Basis of the Birthday Attack

Appendix V Evaluation Criteria for SHA-3

Appendix W The Complexity of Algorithms

Appendix X Radix-64 Conversion

Appendix Y The Base Rate Fallacy

Glossary

NOTATION

Symbol Expression Meaning

D, K D(K, Y) Symmetric decryption of ciphertext Y using secret key K

D, PRa D(PRa, Y) Asymmetric decryption of ciphertext Y using A’s private key PRa

D, PUa D(PUa, Y) Asymmetric decryption of ciphertext Y using A’s public key PUa

E, K E(K, X) Symmetric encryption of plaintext X using secret key K

E, PRa E(PRa, X) Asymmetric encryption of plaintext X using A’s private key PRa

E, PUa E(PUa, X) Asymmetric encryption of plaintext X using A’s public key PUa

K Secret key

PRa Private key of user A

PUa Public key of user A

MAC, K MAC(K, X) Message authentication code of message X using secret key K

GF(p)
The finite field of order p, where p is prime.The field is defined as
the set Zp together with the arithmetic operations modulo p.

GF(2n) The finite field of order 2n

Zn Set of nonnegative integers less than n

gcd gcd(i, j)
Greatest common divisor; the largest positive integer that
divides both i and j with no remainder on division.

mod a mod m Remainder after division of a by m

mod, K a K b (mod m) a mod m = b mod m

mod, [ a [ b (mod m) a mod m ≠ b mod m

dlog dloga,p(b) Discrete logarithm of the number b for the base a (mod p)

w f(n)
The number of positive integers less than n and relatively
prime to n.
This is Euler’s totient function.

Σ a
n

i=1
ai a1 + a2 + g + an

Π q
n

i=1
ai a1 * a2 * g * an

� i � j
i divides j, which means that there is no remainder when j is
divided by i

� , � �a � Absolute value of a

10

Hiva-Network.Com

NOTATION 11

Symbol Expression Meaning

} x } y x concatenated with y

≈ x ≈ y x is approximately equal to y

⊕ x⊕ y Exclusive-OR of x and y for single-bit variables;
Bitwise exclusive-OR of x and y for multiple-bit variables

:, ; :x; The largest integer less than or equal to x
∈ x∈ S The element x is contained in the set S.

·
A · (a1, a2,
c ak)

The integer A corresponds to the sequence of integers
(a1, a2, c ak)

PREFACE

WHAT’S NEW IN THE SEVENTH EDITION

In the four years since the sixth edition of this book was published, the field has seen contin-
ued innovations and improvements. In this new edition, I try to capture these changes while
maintaining a broad and comprehensive coverage of the entire field. To begin this process of
revision, the sixth edition of this book was extensively reviewed by a number of professors
who teach the subject and by professionals working in the field. The result is that, in many
places, the narrative has been clarified and tightened, and illustrations have been improved.

Beyond these refinements to improve pedagogy and user-friendliness, there have been
substantive changes throughout the book. Roughly the same chapter organization has been
retained, but much of the material has been revised and new material has been added. The
most noteworthy changes are as follows:

■ Fundamental security design principles: Chapter 1 includes a new section discussing the
security design principles listed as fundamental by the National Centers of Academic
Excellence in Information Assurance/Cyber Defense, which is jointly sponsored by the
U.S. National Security Agency and the U.S. Department of Homeland Security.

■ Attack surfaces and attack trees: Chapter 1 includes a new section describing these two
concepts, which are useful in evaluating and classifying security threats.

■ Number theory coverage: The material on number theory has been consolidated
into a single chapter, Chapter 2. This makes for a convenient reference. The relevant
portions of Chapter 2 can be assigned as needed.

■ Finite fields: The chapter on finite fields has been revised and expanded with addi-
tional text and new figures to enhance understanding.

■ Format-preserving encryption: This relatively new mode of encryption is enjoying
increasing commercial success. A new section in Chapter 7 covers this method.

■ Conditioning and health testing for true random number generators: Chapter 8 now
provides coverage of these important topics.

■ User authentication model: Chapter 15 includes a new description of a general model
for user authentication, which helps to unify the discussion of the various approaches
to user authentication.

■ Cloud security: The material on cloud security in Chapter 16 has been updated and
expanded to reflect its importance and recent developments.

■ Transport Layer Security (TLS): The treatment of TLS in Chapter 17 has been updated,
reorganized to improve clarity, and now includes a discussion of the new TLS version 1.3.

■ Email Security: Chapter 19 has been completely rewritten to provide a comprehensive
and up-to-date discussion of email security. It includes:

— New: discussion of email threats and a comprehensive approach to email security.

— New: discussion of STARTTLS, which provides confidentiality and authentication
for SMTP.

12

PREFACE 13

— Revised: treatment of S/MIME has been updated to reflect the latest version 3.2.

— New: discussion of DNSSEC and its role in supporting email security.

— New: discussion of DNS-based Authentication of Named Entities (DANE) and the
use of this approach to enhance security for certificate use in SMTP and S/MIME.

— New: discussion of Sender Policy Framework (SPF), which is the standardized way
for a sending domain to identify and assert the mail senders for a given domain.

— Revised: discussion of DomainKeys Identified Mail (DKIM) has been revised.

— New: discussion of Domain-based Message Authentication, Reporting, and Confor-
mance (DMARC) allows email senders to specify policy on how their mail should
be handled, the types of reports that receivers can send back, and the frequency
those reports should be sent.

OBJECTIVES

The subject, and therefore this book, draws on a variety of disciplines. In particular,
it is impossible to appreciate the significance of some of the techniques discussed in this
book without a basic understanding of number theory and some results from probability
theory. Nevertheless, an attempt has been made to make the book self-contained. The book
not only presents the basic mathematical results that are needed but provides the reader
with an intuitive understanding of those results. Such background material is introduced
as needed. This approach helps to motivate the material that is introduced, and the author
considers this preferable to simply presenting all of the mathematical material in a lump at
the beginning of the book.

SUPPORT OF ACM/IEEE COMPUTER SCIENCE CURRICULA 2013

The book is intended for both academic and professional audiences. As a textbook, it is
intended as a one-semester undergraduate course in cryptography and network security for
computer science, computer engineering, and electrical engineering majors. The changes to
this edition are intended to provide support of the ACM/IEEE Computer Science Curricula
2013 (CS2013). CS2013 adds Information Assurance and Security (IAS) to the curriculum rec-
ommendation as one of the Knowledge Areas in the Computer Science Body of Knowledge.
The document states that IAS is now part of the curriculum recommendation because of the
critical role of IAS in computer science education. CS2013 divides all course work into three
categories: Core-Tier 1 (all topics should be included in the curriculum), Core-Tier-2 (all or
almost all topics should be included), and elective (desirable to provide breadth and depth).
In the IAS area, CS2013 recommends topics in Fundamental Concepts and Network Security

It is the purpose of this book to provide a practical survey of both the principles and practice
of cryptography and network security. In the first part of the book, the basic issues to be
addressed by a network security capability are explored by providing a tutorial and survey
of cryptography and network security technology. The latter part of the book deals with the
practice of network security: practical applications that have been implemented and are in
use to provide network security.

14 PREFACE

in Tier 1 and Tier 2, and Cryptography topics as elective. This text covers virtually all of the
topics listed by CS2013 in these three categories.

The book also serves as a basic reference volume and is suitable for self-study.

PLAN OF THE TEXT

The book is divided into eight parts.

■ Background

■ Symmetric Ciphers

■ Asymmetric Ciphers

■ Cryptographic Data Integrity Algorithms

■ Mutual Trust

■ Network and Internet Security

■ System Security

■ Legal and Ethical Issues

The book includes a number of pedagogic features, including the use of the computer
algebra system Sage and numerous figures and tables to clarify the discussions. Each chap-
ter includes a list of key words, review questions, homework problems, and suggestions
for further reading. The book also includes an extensive glossary, a list of frequently used
acronyms, and a bibliography. In addition, a test bank is available to instructors.

INSTRUCTOR SUPPORT MATERIALS

The major goal of this text is to make it as effective a teaching tool for this exciting and
fast-moving subject as possible. This goal is reflected both in the structure of the book and in
the supporting material. The text is accompanied by the following supplementary material
that will aid the instructor:

■ Solutions manual: Solutions to all end-of-chapter Review Questions and Problems.

■ Projects manual: Suggested project assignments for all of the project categories listed
below.

■ PowerPoint slides: A set of slides covering all chapters, suitable for use in lecturing.

■ PDF files: Reproductions of all figures and tables from the book.

■ Test bank: A chapter-by-chapter set of questions with a separate file of answers.

■ Sample syllabuses: The text contains more material than can be conveniently covered
in one semester. Accordingly, instructors are provided with several sample syllabuses
that guide the use of the text within limited time.

All of these support materials are available at the Instructor Resource Center
(IRC) for this textbook, which can be reached through the publisher’s Web site
www.pearsonglobaleditions.com/stallings. To gain access to the IRC, please contact your
local Pearson sales representative.

PREFACE 15

PROJECTS AND OTHER STUDENT EXERCISES

For many instructors, an important component of a cryptography or network security course
is a project or set of projects by which the student gets hands-on experience to reinforce
concepts from the text. This book provides an unparalleled degree of support, including a
projects component in the course. The IRC not only includes guidance on how to assign and
structure the projects, but also includes a set of project assignments that covers a broad range
of topics from the text:

■ Sage projects: Described in the next section.

■ Hacking project: Exercise designed to illuminate the key issues in intrusion detection
and prevention.

■ Block cipher projects: A lab that explores the operation of the AES encryption algo-
rithm by tracing its execution, computing one round by hand, and then exploring the
various block cipher modes of use. The lab also covers DES. In both cases, an online
Java applet is used (or can be downloaded) to execute AES or DES.

■ Lab exercises: A series of projects that involve programming and experimenting with
concepts from the book.

■ Research projects: A series of research assignments that instruct the student to research
a particular topic on the Internet and write a report.

■ Programming projects: A series of programming projects that cover a broad range of
topics and that can be implemented in any suitable language on any platform.

■ Practical security assessments: A set of exercises to examine current infrastructure and
practices of an existing organization.

■ Firewall projects: A portable network firewall visualization simulator, together with
exercises for teaching the fundamentals of firewalls.

■ Case studies: A set of real-world case studies, including learning objectives, case
description, and a series of case discussion questions.

■ Writing assignments: A set of suggested writing assignments, organized by chapter.

■ Reading/report assignments: A list of papers in the literature—one for each chapter—
that can be assigned for the student to read and then write a short report.

This diverse set of projects and other student exercises enables the instructor to use
the book as one component in a rich and varied learning experience and to tailor a course
plan to meet the specific needs of the instructor and students. See Appendix A in this book
for details.

THE SAGE COMPUTER ALGEBRA SYSTEM

One of the most important features of this book is the use of Sage for cryptographic examples
and homework assignments. Sage is an open-source, multiplatform, freeware package that
implements a very powerful, flexible, and easily learned mathematics and computer algebra
system. Unlike competing systems (such as Mathematica, Maple, and MATLAB), there are

16 PREFACE

no licensing agreements or fees involved. Thus, Sage can be made available on computers
and networks at school, and students can individually download the software to their own
personal computers for use at home. Another advantage of using Sage is that students learn
a powerful, flexible tool that can be used for virtually any mathematical application, not
just cryptography.

The use of Sage can make a significant difference to the teaching of the mathematics
of cryptographic algorithms. This book provides a large number of examples of the use of
Sage covering many cryptographic concepts in Appendix B, which is included in this book.

Appendix C lists exercises in each of these topic areas to enable the student to gain
hands-on experience with cryptographic algorithms. This appendix is available to instruc-
tors at the IRC for this book. Appendix C includes a section on how to download and get
started with Sage, a section on programming with Sage, and exercises that can be assigned to
students in the following categories:

■ Chapter 2—Number Theory and Finite Fields: Euclidean and extended Euclidean
algorithms, polynomial arithmetic, GF(24), Euler’s Totient function, Miller–Rabin, fac-
toring, modular exponentiation, discrete logarithm, and Chinese remainder theorem.

■ Chapter 3—Classical Encryption: Affine ciphers and the Hill cipher.

■ Chapter 4—Block Ciphers and the Data Encryption Standard: Exercises based
on SDES.

■ Chapter 6—Advanced Encryption Standard: Exercises based on SAES.

■ Chapter 8—Pseudorandom Number Generation and Stream Ciphers: Blum Blum
Shub, linear congruential generator, and ANSI X9.17 PRNG.

■ Chapter 9—Public-Key Cryptography and RSA: RSA encrypt/decrypt and signing.

■ Chapter 10—Other Public-Key Cryptosystems: Diffie–Hellman, elliptic curve.

■ Chapter 11—Cryptographic Hash Functions: Number-theoretic hash function.

■ Chapter 13—Digital Signatures: DSA.

ONLINE DOCUMENTS FOR STUDENTS

For this new edition, a tremendous amount of original supporting material for students has
been made available online.

Purchasing this textbook new also grants the reader six months of access to the
Companion Website, which includes the following materials:

■ Online chapters: To limit the size and cost of the book, four chapters of the book are
provided in PDF format. This includes three chapters on computer security and one on
legal and ethical issues. The chapters are listed in this book’s table of contents.

■ Online appendices: There are numerous interesting topics that support material found
in the text but whose inclusion is not warranted in the printed text. A total of 20 online
appendices cover these topics for the interested student. The appendices are listed in
this book’s table of contents.

PREFACE 17

■ Homework problems and solutions: To aid the student in understanding the material,
a separate set of homework problems with solutions are available.

■ Key papers: A number of papers from the professional literature, many hard to find,
are provided for further reading.

■ Supporting documents: A variety of other useful documents are referenced in the text
and provided online.

■ Sage code: The Sage code from the examples in Appendix B is useful in case the student
wants to play around with the examples.

To access the Companion Website, follow the instructions for “digital resources for
students” found in the front of this book.

ACKNOWLEDGMENTS

This new edition has benefited from review by a number of people who gave generously
of their time and expertise. The following professors reviewed all or a large part of the
manuscript: Hossein Beyzavi (Marymount University), Donald F. Costello (University of
Nebraska–Lincoln), James Haralambides (Barry University), Anand Seetharam (California
State University at Monterey Bay), Marius C. Silaghi (Florida Institute of Technology),
Shambhu Upadhyaya (University at Buffalo), Zhengping Wu (California State University
at San Bernardino), Liangliang Xiao (Frostburg State University), Seong-Moo (Sam) Yoo
(The University of Alabama in Huntsville), and Hong Zhang (Armstrong State University).

Thanks also to the people who provided detailed technical reviews of one or more
chapters: Dino M. Amaral, Chris Andrew, Prof. (Dr). C. Annamalai, Andrew Bain, Riccardo
Bernardini, Olivier Blazy, Zervopoulou Christina, Maria Christofi, Dhananjoy Dey, Mario
Emmanuel, Mike Fikuart, Alexander Fries, Pierpaolo Giacomin, Pedro R. M. Inácio,
Daniela Tamy Iwassa, Krzysztof Janowski, Sergey Katsev, Adnan Kilic, Rob Knox, Mina
Pourdashty, Yuri Poeluev, Pritesh Prajapati, Venkatesh Ramamoorthy, Andrea Razzini,
Rami Rosen, Javier Scodelaro, Jamshid Shokrollahi, Oscar So, and David Tillemans.

In addition, I was fortunate to have reviews of individual topics by “subject-area
gurus,” including Jesse Walker of Intel (Intel’s Digital Random Number Generator), Russ
Housley of Vigil Security (key wrapping), Joan Daemen (AES), Edward F. Schaefer of
Santa Clara University (Simplified AES), Tim Mathews, formerly of RSA Laboratories
(S/MIME), Alfred Menezes of the University of Waterloo (elliptic curve cryptography),
William Sutton, Editor/Publisher of The Cryptogram (classical encryption), Avi Rubin of
Johns Hopkins University (number theory), Michael Markowitz of Information Security
Corporation (SHA and DSS), Don Davis of IBM Internet Security Systems (Kerberos),
Steve Kent of BBN Technologies (X.509), and Phil Zimmerman (PGP).

Nikhil Bhargava (IIT Delhi) developed the set of online homework problems and
solutions. Dan Shumow of Microsoft and the University of Washington developed all of
the Sage examples and assignments in Appendices B and C. Professor Sreekanth Malladi of
Dakota State University developed the hacking exercises. Lawrie Brown of the Australian
Defence Force Academy provided the AES/DES block cipher projects and the security
assessment assignments.

18 PREFACE

Sanjay Rao and Ruben Torres of Purdue University developed the laboratory exercises
that appear in the IRC. The following people contributed project assignments that appear in
the instructor’s supplement: Henning Schulzrinne (Columbia University); Cetin Kaya Koc
(Oregon State University); and David Balenson (Trusted Information Systems and George
Washington University). Kim McLaughlin developed the test bank.

Finally, I thank the many people responsible for the publication of this book, all of
whom did their usual excellent job. This includes the staff at Pearson, particularly my editor
Tracy Johnson, program manager Carole Snyder, and production manager Bob Engelhardt.
Thanks also to the marketing and sales staffs at Pearson, without whose efforts this book
would not be in front of you.

ACKNOWLEDGMENTS FOR THE GLOBAL EDITION

Pearson would like to thank and acknowledge Somitra Kumar Sanadhya (Indraprastha
Institute of Information Technology Delhi), and Somanath Tripathy (Indian Institute of
Technology Patna) for contributing to the Global Edition, and Anwitaman Datta (Nanyang
Technological University Singapore), Atul Kahate (Pune University), Goutam Paul (Indian
Statistical Institute Kolkata), and Khyat Sharma for reviewing the Global Edition.

ABOUT THE AUTHOR

Dr. William Stallings has authored 18 titles, and counting revised editions, over 40 books
on computer security, computer networking, and computer architecture. His writings have
appeared in numerous publications, including the Proceedings of the IEEE, ACM Computing
Reviews, and Cryptologia.

He has 13 times received the award for the best Computer Science textbook of the
year from the Text and Academic Authors Association.

In over 30 years in the field, he has been a technical contributor, technical manager,
and an executive with several high-technology firms. He has designed and implemented
both TCP/IP-based and OSI-based protocol suites on a variety of computers and operating
systems, ranging from microcomputers to mainframes. As a consultant, he has advised gov-
ernment agencies, computer and software vendors, and major users on the design, selection,
and use of networking software and products.

He created and maintains the Computer Science Student Resource Site at
ComputerScienceStudent.com. This site provides documents and links on a variety of
subjects of general interest to computer science students (and professionals). He is a member
of the editorial board of Cryptologia, a scholarly journal devoted to all aspects of cryptology.

Dr. Stallings holds a PhD from MIT in computer science and a BS from Notre Dame
in electrical engineering.

19

PART ONE: BACKGROUND

CHAPTER

Computer and Network
Security Concepts

1.1 Computer Security Concepts

A Definition of Computer Security
Examples
The Challenges of Computer Security

1.2 The OSI Security Architecture

1.3 Security Attacks

Passive Attacks
Active Attacks

1.4 Security Services

Authentication
Access Control
Data Confidentiality
Data Integrity
Nonrepudiation
Availability Service

1.5 Security Mechanisms

1.6 Fundamental Security Design Principles

1.7 Attack Surfaces and Attack Trees

Attack Surfaces
Attack Trees

1.8 A Model for Network Security

1.9 Standards

1.10 Key Terms, Review Questions, and Problems

19

Hiva-Network.Com

20 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

This book focuses on two broad areas: cryptographic algorithms and protocols, which
have a broad range of applications; and network and Internet security, which rely
heavily on cryptographic techniques.

Cryptographic algorithms and protocols can be grouped into four main areas:

■ Symmetric encryption: Used to conceal the contents of blocks or streams of
data of any size, including messages, files, encryption keys, and passwords.

■ Asymmetric encryption: Used to conceal small blocks of data, such as encryp-
tion keys and hash function values, which are used in digital signatures.

■ Data integrity algorithms: Used to protect blocks of data, such as messages,
from alteration.

■ Authentication protocols: These are schemes based on the use of crypto-
graphic algorithms designed to authenticate the identity of entities.

The field of network and Internet security consists of measures to deter, prevent,
detect, and correct security violations that involve the transmission of information.
That is a broad statement that covers a host of possibilities. To give you a feel for the
areas covered in this book, consider the following examples of security violations:

1. User A transmits a file to user B. The file contains sensitive information
(e.g., payroll records) that is to be protected from disclosure. User C, who is
not authorized to read the file, is able to monitor the transmission and capture
a copy of the file during its transmission.

2. A network manager, D, transmits a message to a computer, E, under its man-
agement. The message instructs computer E to update an authorization file to
include the identities of a number of new users who are to be given access to
that computer. User F intercepts the message, alters its contents to add or delete
entries, and then forwards the message to computer E, which accepts the mes-
sage as coming from manager D and updates its authorization file accordingly.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Describe the key security requirements of confidentiality, integrity, and
availability.

◆ Describe the X.800 security architecture for OSI.

◆ Discuss the types of security threats and attacks that must be dealt with
and give examples of the types of threats and attacks that apply to differ-
ent categories of computer and network assets.

◆ Explain the fundamental security design principles.

◆ Discuss the use of attack surfaces and attack trees.

◆ List and briefly describe key organizations involved in cryptography
standards.

1.1 / COMPUTER SECURITY CONCEPTS 21

3. Rather than intercept a message, user F constructs its own message with the
desired entries and transmits that message to computer E as if it had come
from manager D. Computer E accepts the message as coming from manager D
and updates its authorization file accordingly.

4. An employee is fired without warning. The personnel manager sends a mes-
sage to a server system to invalidate the employee’s account. When the invali-
dation is accomplished, the server is to post a notice to the employee’s file as
confirmation of the action. The employee is able to intercept the message and
delay it long enough to make a final access to the server to retrieve sensitive
information. The message is then forwarded, the action taken, and the confir-
mation posted. The employee’s action may go unnoticed for some consider-
able time.

5. A message is sent from a customer to a stockbroker with instructions for vari-
ous transactions. Subsequently, the investments lose value and the customer
denies sending the message.

Although this list by no means exhausts the possible types of network security viola-
tions, it illustrates the range of concerns of network security.

1.1 COMPUTER SECURITY CONCEPTS

A Definition of Computer Security

The NIST Computer Security Handbook [NIST95] defines the term computer secu-
rity as follows:

Computer Security: The protection afforded to an automated information system
in order to attain the applicable objectives of preserving the integrity, availability,
and confidentiality of information system resources (includes hardware, software,
firmware, information/data, and telecommunications).

This definition introduces three key objectives that are at the heart of com-
puter security:

■ Confidentiality: This term covers two related concepts:

Data1 confidentiality: Assures that private or confidential information is
not made available or disclosed to unauthorized individuals.

Privacy: Assures that individuals control or influence what information re-
lated to them may be collected and stored and by whom and to whom that
information may be disclosed.

1RFC 4949 defines information as “facts and ideas, which can be represented (encoded) as various forms
of data,” and data as “information in a specific physical representation, usually a sequence of symbols
that have meaning; especially a representation of information that can be processed or produced by a
computer.” Security literature typically does not make much of a distinction, nor does this book.

22 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

■ Integrity: This term covers two related concepts:

Data integrity: Assures that information (both stored and in transmit-
ted packets) and programs are changed only in a specified and authorized
manner.

System integrity: Assures that a system performs its intended function in
an unimpaired manner, free from deliberate or inadvertent unauthorized
manipulation of the system.

■ Availability: Assures that systems work promptly and service is not denied to
authorized users.

These three concepts form what is often referred to as the CIA triad. The
three concepts embody the fundamental security objectives for both data and for
information and computing services. For example, the NIST standard FIPS 199
(Standards for Security Categorization of Federal Information and Information
Systems) lists confidentiality, integrity, and availability as the three security objec-
tives for information and for information systems. FIPS 199 provides a useful char-
acterization of these three objectives in terms of requirements and the definition of
a loss of security in each category:

■ Confidentiality: Preserving authorized restrictions on information access
and disclosure, including means for protecting personal privacy and propri-
etary information. A loss of confidentiality is the unauthorized disclosure of
information.

■ Integrity: Guarding against improper information modification or destruc-
tion, including ensuring information nonrepudiation and authenticity. A loss
of integrity is the unauthorized modification or destruction of information.

■ Availability: Ensuring timely and reliable access to and use of information.
A loss of availability is the disruption of access to or use of information or an
information system.

Although the use of the CIA triad to define security objectives is well estab-
lished, some in the security field feel that additional concepts are needed to present a
complete picture (Figure 1.1). Two of the most commonly mentioned are as follows:

Figure 1.1 Essential Network and Computer Security
Requirements

Data
and

services

Availability

Integrity

A
ccountability

A
ut

he
nt

ic
ity

Co
nfid

ent
iali

ty

1.1 / COMPUTER SECURITY CONCEPTS 23

■ Authenticity: The property of being genuine and being able to be verified and
trusted; confidence in the validity of a transmission, a message, or message
originator. This means verifying that users are who they say they are and that
each input arriving at the system came from a trusted source.

■ Accountability: The security goal that generates the requirement for actions
of an entity to be traced uniquely to that entity. This supports nonrepudia-
tion, deterrence, fault isolation, intrusion detection and prevention, and after-
action recovery and legal action. Because truly secure systems are not yet an
achievable goal, we must be able to trace a security breach to a responsible
party. Systems must keep records of their activities to permit later forensic
analysis to trace security breaches or to aid in transaction disputes.

Examples

We now provide some examples of applications that illustrate the requirements just
enumerated.2 For these examples, we use three levels of impact on organizations or
individuals should there be a breach of security (i.e., a loss of confidentiality, integ-
rity, or availability). These levels are defined in FIPS PUB 199:

■ Low: The loss could be expected to have a limited adverse effect on organi-
zational operations, organizational assets, or individuals. A limited adverse
effect means that, for example, the loss of confidentiality, integrity, or avail-
ability might (i) cause a degradation in mission capability to an extent and
duration that the organization is able to perform its primary functions, but the
effectiveness of the functions is noticeably reduced; (ii) result in minor dam-
age to organizational assets; (iii) result in minor financial loss; or (iv) result in
minor harm to individuals.

■ Moderate: The loss could be expected to have a serious adverse effect on
organizational operations, organizational assets, or individuals. A serious
adverse effect means that, for example, the loss might (i) cause a signifi-
cant degradation in mission capability to an extent and duration that the
organization is able to perform its primary functions, but the effectiveness
of the functions is significantly reduced; (ii) result in significant damage to
organizational assets; (iii) result in significant financial loss; or (iv) result in
significant harm to individuals that does not involve loss of life or serious,
life-threatening injuries.

■ High: The loss could be expected to have a severe or catastrophic adverse
effect on organizational operations, organizational assets, or individuals.
A severe or catastrophic adverse effect means that, for example, the loss
might (i) cause a severe degradation in or loss of mission capability to an
extent and duration that the organization is not able to perform one or more
of its primary functions; (ii) result in major damage to organizational assets;
(iii) result in major financial loss; or (iv) result in severe or catastrophic harm
to individuals involving loss of life or serious, life-threatening injuries.

2These examples are taken from a security policy document published by the Information Technology
Security and Privacy Office at Purdue University.

24 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

CONFIDENTIALITY Student grade information is an asset whose confidentiality is
considered to be highly important by students. In the United States, the release of
such information is regulated by the Family Educational Rights and Privacy Act
(FERPA). Grade information should only be available to students, their parents,
and employees that require the information to do their job. Student enrollment
information may have a moderate confidentiality rating. While still covered by
FERPA, this information is seen by more people on a daily basis, is less likely to be
targeted than grade information, and results in less damage if disclosed. Directory
information, such as lists of students or faculty or departmental lists, may be as-
signed a low confidentiality rating or indeed no rating. This information is typically
freely available to the public and published on a school’s Web site.

INTEGRITY Several aspects of integrity are illustrated by the example of a hospital
patient’s allergy information stored in a database. The doctor should be able to
trust that the information is correct and current. Now suppose that an employee
(e.g., a nurse) who is authorized to view and update this information deliberately
falsifies the data to cause harm to the hospital. The database needs to be restored
to a trusted basis quickly, and it should be possible to trace the error back to the
person responsible. Patient allergy information is an example of an asset with a high
requirement for integrity. Inaccurate information could result in serious harm or
death to a patient and expose the hospital to massive liability.

An example of an asset that may be assigned a moderate level of integrity
requirement is a Web site that offers a forum to registered users to discuss some
specific topic. Either a registered user or a hacker could falsify some entries or
deface the Web site. If the forum exists only for the enjoyment of the users, brings
in little or no advertising revenue, and is not used for something important such
as research, then potential damage is not severe. The Web master may experience
some data, financial, and time loss.

An example of a low integrity requirement is an anonymous online poll. Many
Web sites, such as news organizations, offer these polls to their users with very few
safeguards. However, the inaccuracy and unscientific nature of such polls is well
understood.

AVAILABILITY The more critical a component or service, the higher is the level of
availability required. Consider a system that provides authentication services for
critical systems, applications, and devices. An interruption of service results in the
inability for customers to access computing resources and staff to access the re-
sources they need to perform critical tasks. The loss of the service translates into a
large financial loss in lost employee productivity and potential customer loss.

An example of an asset that would typically be rated as having a moderate
availability requirement is a public Web site for a university; the Web site provides
information for current and prospective students and donors. Such a site is not a
critical component of the university’s information system, but its unavailability will
cause some embarrassment.

An online telephone directory lookup application would be classified as a low
availability requirement. Although the temporary loss of the application may be
an annoyance, there are other ways to access the information, such as a hardcopy
directory or the operator.

1.1 / COMPUTER SECURITY CONCEPTS 25

The Challenges of Computer Security

Computer and network security is both fascinating and complex. Some of the
reasons follow:

1. Security is not as simple as it might first appear to the novice. The require-
ments seem to be straightforward; indeed, most of the major requirements for
security services can be given self-explanatory, one-word labels: confidential-
ity, authentication, nonrepudiation, or integrity. But the mechanisms used to
meet those requirements can be quite complex, and understanding them may
involve rather subtle reasoning.

2. In developing a particular security mechanism or algorithm, one must always
consider potential attacks on those security features. In many cases, successful
attacks are designed by looking at the problem in a completely different way,
therefore exploiting an unexpected weakness in the mechanism.

3. Because of point 2, the procedures used to provide particular services are
often counterintuitive. Typically, a security mechanism is complex, and it is not
obvious from the statement of a particular requirement that such elaborate
measures are needed. It is only when the various aspects of the threat are con-
sidered that elaborate security mechanisms make sense.

4. Having designed various security mechanisms, it is necessary to decide where
to use them. This is true both in terms of physical placement (e.g., at what points
in a network are certain security mechanisms needed) and in a logical sense
(e.g., at what layer or layers of an architecture such as TCP/IP [Transmission
Control Protocol/Internet Protocol] should mechanisms be placed).

5. Security mechanisms typically involve more than a particular algorithm or
protocol. They also require that participants be in possession of some secret in-
formation (e.g., an encryption key), which raises questions about the creation,
distribution, and protection of that secret information. There also may be a re-
liance on communications protocols whose behavior may complicate the task
of developing the security mechanism. For example, if the proper functioning
of the security mechanism requires setting time limits on the transit time of a
message from sender to receiver, then any protocol or network that introduces
variable, unpredictable delays may render such time limits meaningless.

6. Computer and network security is essentially a battle of wits between a per-
petrator who tries to find holes and the designer or administrator who tries to
close them. The great advantage that the attacker has is that he or she need
only find a single weakness, while the designer must find and eliminate all
weaknesses to achieve perfect security.

7. There is a natural tendency on the part of users and system managers to per-
ceive little benefit from security investment until a security failure occurs.

8. Security requires regular, even constant, monitoring, and this is difficult in
today’s short-term, overloaded environment.

9. Security is still too often an afterthought to be incorporated into a system
after the design is complete rather than being an integral part of the design
process.

26 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

10. Many users and even security administrators view strong security as an
impediment to efficient and user-friendly operation of an information system
or use of information.

The difficulties just enumerated will be encountered in numerous ways as we
examine the various security threats and mechanisms throughout this book.

1.2 THE OSI SECURITY ARCHITECTURE

To assess effectively the security needs of an organization and to evaluate and
choose various security products and policies, the manager responsible for security
needs some systematic way of defining the requirements for security and character-
izing the approaches to satisfying those requirements. This is difficult enough in a
centralized data processing environment; with the use of local and wide area net-
works, the problems are compounded.

ITU-T3 Recommendation X.800, Security Architecture for OSI, defines such a
systematic approach.4 The OSI security architecture is useful to managers as a way
of organizing the task of providing security. Furthermore, because this architecture
was developed as an international standard, computer and communications vendors
have developed security features for their products and services that relate to this
structured definition of services and mechanisms.

For our purposes, the OSI security architecture provides a useful, if abstract,
overview of many of the concepts that this book deals with. The OSI security archi-
tecture focuses on security attacks, mechanisms, and services. These can be defined
briefly as

■ Security attack: Any action that compromises the security of information
owned by an organization.

■ Security mechanism: A process (or a device incorporating such a process)
that is designed to detect, prevent, or recover from a security attack.

■ Security service: A processing or communication service that enhances the
security of the data processing systems and the information transfers of an
organization. The services are intended to counter security attacks, and they
make use of one or more security mechanisms to provide the service.

In the literature, the terms threat and attack are commonly used to mean more
or less the same thing. Table 1.1 provides definitions taken from RFC 4949, Internet
Security Glossary.

3The International Telecommunication Union (ITU) Telecommunication Standardization Sector (ITU-T)
is a United Nations-sponsored agency that develops standards, called Recommendations, relating to tele-
communications and to open systems interconnection (OSI).
4The OSI security architecture was developed in the context of the OSI protocol architecture, which is
described in Appendix L. However, for our purposes in this chapter, an understanding of the OSI proto-
col architecture is not required.

1.3 / SECURITY ATTACKS 27

1.3 SECURITY ATTACKS

A useful means of classifying security attacks, used both in X.800 and RFC 4949, is
in terms of passive attacks and active attacks (Figure 1.2). A passive attack attempts
to learn or make use of information from the system but does not affect system re-
sources. An active attack attempts to alter system resources or affect their operation.

Passive Attacks

Passive attacks (Figure 1.2a) are in the nature of eavesdropping on, or monitoring
of, transmissions. The goal of the opponent is to obtain information that is being
transmitted. Two types of passive attacks are the release of message contents and
traffic analysis.

The release of message contents is easily understood. A telephone conver-
sation, an electronic mail message, and a transferred file may contain sensitive or
confidential information. We would like to prevent an opponent from learning the
contents of these transmissions.

A second type of passive attack, traffic analysis, is subtler. Suppose that we
had a way of masking the contents of messages or other information traffic so that
opponents, even if they captured the message, could not extract the information
from the message. The common technique for masking contents is encryption. If we
had encryption protection in place, an opponent might still be able to observe the
pattern of these messages. The opponent could determine the location and identity
of communicating hosts and could observe the frequency and length of messages
being exchanged. This information might be useful in guessing the nature of the
communication that was taking place.

Passive attacks are very difficult to detect, because they do not involve any
alteration of the data. Typically, the message traffic is sent and received in an appar-
ently normal fashion, and neither the sender nor receiver is aware that a third party
has read the messages or observed the traffic pattern. However, it is feasible to pre-
vent the success of these attacks, usually by means of encryption. Thus, the empha-
sis in dealing with passive attacks is on prevention rather than detection.

Active Attacks

Active attacks (Figure 1.2b) involve some modification of the data stream or the
creation of a false stream and can be subdivided into four categories: masquerade,
replay, modification of messages, and denial of service.

Threat
A potential for violation of security, which exists when there is a circumstance, capability, action,
or event that could breach security and cause harm. That is, a threat is a possible danger that might
exploit a vulnerability.

Attack
An assault on system security that derives from an intelligent threat; that is, an intelligent act that
is a deliberate attempt (especially in the sense of a method or technique) to evade security services
and violate the security policy of a system.

Table 1.1 Threats and Attacks (RFC 4949)

28 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

A masquerade takes place when one entity pretends to be a different entity
(path 2 of Figure 1.2b is active). A masquerade attack usually includes one of the
other forms of active attack. For example, authentication sequences can be captured
and replayed after a valid authentication sequence has taken place, thus enabling an
authorized entity with few privileges to obtain extra privileges by impersonating an
entity that has those privileges.

Replay involves the passive capture of a data unit and its subsequent retrans-
mission to produce an unauthorized effect (paths 1, 2, and 3 active).

Modification of messages simply means that some portion of a legitimate mes-
sage is altered, or that messages are delayed or reordered, to produce an unauthor-
ized effect (paths 1 and 2 active). For example, a message meaning “Allow John
Smith to read confidential file accounts” is modified to mean “Allow Fred Brown to
read confidential file accounts.”

Figure 1.2 Security Attacks

(a) Passive attacks

Alice

(b) Active attacks

Bob

Darth

Bob

Darth

Alice

Internet or
other communications facility

Internet or
other communications facility

1 2
3

Hiva-Network.Com

1.4 / SECURITY SERVICES 29

The denial of service prevents or inhibits the normal use or management of
communications facilities (path 3 active). This attack may have a specific target; for
example, an entity may suppress all messages directed to a particular destination
(e.g., the security audit service). Another form of service denial is the disruption of
an entire network, either by disabling the network or by overloading it with mes-
sages so as to degrade performance.

Active attacks present the opposite characteristics of passive attacks. Whereas
passive attacks are difficult to detect, measures are available to prevent their success.
On the other hand, it is quite difficult to prevent active attacks absolutely because
of the wide variety of potential physical, software, and network vulnerabilities.
Instead, the goal is to detect active attacks and to recover from any disruption or
delays caused by them. If the detection has a deterrent effect, it may also contribute
to prevention.

1.4 SECURITY SERVICES

X.800 defines a security service as a service that is provided by a protocol layer of
communicating open systems and that ensures adequate security of the systems or
of data transfers. Perhaps a clearer definition is found in RFC 4949, which provides
the following definition: a processing or communication service that is provided by
a system to give a specific kind of protection to system resources; security services
implement security policies and are implemented by security mechanisms.

X.800 divides these services into five categories and fourteen specific services
(Table 1.2). We look at each category in turn.5

Authentication

The authentication service is concerned with assuring that a communication is au-
thentic. In the case of a single message, such as a warning or alarm signal, the function
of the authentication service is to assure the recipient that the message is from the
source that it claims to be from. In the case of an ongoing interaction, such as the con-
nection of a terminal to a host, two aspects are involved. First, at the time of connec-
tion initiation, the service assures that the two entities are authentic, that is, that each
is the entity that it claims to be. Second, the service must assure that the connection is
not interfered with in such a way that a third party can masquerade as one of the two
legitimate parties for the purposes of unauthorized transmission or reception.

Two specific authentication services are defined in X.800:

■ Peer entity authentication: Provides for the corroboration of the identity of a
peer entity in an association. Two entities are considered peers if they imple-
ment to same protocol in different systems; for example two TCP modules
in two communicating systems. Peer entity authentication is provided for

5There is no universal agreement about many of the terms used in the security literature. For example, the
term integrity is sometimes used to refer to all aspects of information security. The term authentication is
sometimes used to refer both to verification of identity and to the various functions listed under integrity
in this chapter. Our usage here agrees with both X.800 and RFC 4949.

30 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

AUTHENTICATION

The assurance that the communicating entity is the
one that it claims to be.

Peer Entity Authentication
Used in association with a logical connection to
provide confidence in the identity of the entities
connected.

Data-Origin Authentication
In a connectionless transfer, provides assurance that
the source of received data is as claimed.

ACCESS CONTROL

The prevention of unauthorized use of a resource
(i.e., this service controls who can have access to a
resource, under what conditions access can occur,
and what those accessing the resource are allowed
to do).

DATA CONFIDENTIALITY

The protection of data from unauthorized
disclosure.

Connection Confidentiality
The protection of all user data on a connection.

Connectionless Confidentiality
The protection of all user data in a single data block.

Selective-Field Confidentiality
The confidentiality of selected fields within the user
data on a connection or in a single data block.

Traffic-Flow Confidentiality
The protection of the information that might be
derived from observation of traffic flows.

DATA INTEGRITY

The assurance that data received are exactly as
sent by an authorized entity (i.e., contain no modi-
fication, insertion, deletion, or replay).

Connection Integrity with Recovery
Provides for the integrity of all user data on a connec-
tion and detects any modification, insertion, deletion,
or replay of any data within an entire data sequence,
with recovery attempted.

Connection Integrity without Recovery
As above, but provides only detection without
recovery.

Selective-Field Connection Integrity
Provides for the integrity of selected fields within the
user data of a data block transferred over a connec-
tion and takes the form of determination of whether
the selected fields have been modified, inserted,
deleted, or replayed.

Connectionless Integrity
Provides for the integrity of a single connectionless
data block and may take the form of detection of
data modification. Additionally, a limited form of
replay detection may be provided.

Selective-Field Connectionless Integrity
Provides for the integrity of selected fields within a
single connectionless data block; takes the form of
determination of whether the selected fields have
been modified.

NONREPUDIATION

Provides protection against denial by one of the
entities involved in a communication of having par-
ticipated in all or part of the communication.

Nonrepudiation, Origin
Proof that the message was sent by the specified
party.

Nonrepudiation, Destination
Proof that the message was received by the specified
party.

Table 1.2 Security Services (X.800)

use at the establishment of, or at times during the data transfer phase of, a
connection. It attempts to provide confidence that an entity is not performing
either a masquerade or an unauthorized replay of a previous connection.

■ Data origin authentication: Provides for the corroboration of the source of a
data unit. It does not provide protection against the duplication or modifica-
tion of data units. This type of service supports applications like electronic mail,
where there are no prior interactions between the communicating entities.

1.4 / SECURITY SERVICES 31

Access Control

In the context of network security, access control is the ability to limit and control
the access to host systems and applications via communications links. To achieve
this, each entity trying to gain access must first be identified, or authenticated,
so that access rights can be tailored to the individual.

Data Confidentiality

Confidentiality is the protection of transmitted data from passive attacks. With re-
spect to the content of a data transmission, several levels of protection can be iden-
tified. The broadest service protects all user data transmitted between two users
over a period of time. For example, when a TCP connection is set up between two
systems, this broad protection prevents the release of any user data transmitted over
the TCP connection. Narrower forms of this service can also be defined, including
the protection of a single message or even specific fields within a message. These
refinements are less useful than the broad approach and may even be more complex
and expensive to implement.

The other aspect of confidentiality is the protection of traffic flow from
analysis. This requires that an attacker not be able to observe the source and desti-
nation, frequency, length, or other characteristics of the traffic on a communications
facility.

Data Integrity

As with confidentiality, integrity can apply to a stream of messages, a single mes-
sage, or selected fields within a message. Again, the most useful and straightforward
approach is total stream protection.

A connection-oriented integrity service, one that deals with a stream of mes-
sages, assures that messages are received as sent with no duplication, insertion,
modification, reordering, or replays. The destruction of data is also covered under
this service. Thus, the connection-oriented integrity service addresses both mes-
sage stream modification and denial of service. On the other hand, a connection-
less integrity service, one that deals with individual messages without regard to any
larger context, generally provides protection against message modification only.

We can make a distinction between service with and without recovery. Because
the integrity service relates to active attacks, we are concerned with detection rather
than prevention. If a violation of integrity is detected, then the service may simply
report this violation, and some other portion of software or human intervention is
required to recover from the violation. Alternatively, there are mechanisms avail-
able to recover from the loss of integrity of data, as we will review subsequently. The
incorporation of automated recovery mechanisms is, in general, the more attractive
alternative.

Nonrepudiation

Nonrepudiation prevents either sender or receiver from denying a transmitted mes-
sage. Thus, when a message is sent, the receiver can prove that the alleged sender in
fact sent the message. Similarly, when a message is received, the sender can prove
that the alleged receiver in fact received the message.

32 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

Availability Service

Both X.800 and RFC 4949 define availability to be the property of a system or a
system resource being accessible and usable upon demand by an authorized system
entity, according to performance specifications for the system (i.e., a system is avail-
able if it provides services according to the system design whenever users request
them). A variety of attacks can result in the loss of or reduction in availability. Some
of these attacks are amenable to automated countermeasures, such as authentica-
tion and encryption, whereas others require some sort of physical action to prevent
or recover from loss of availability of elements of a distributed system.

X.800 treats availability as a property to be associated with various security
services. However, it makes sense to call out specifically an availability service. An
availability service is one that protects a system to ensure its availability. This ser-
vice addresses the security concerns raised by denial-of-service attacks. It depends
on proper management and control of system resources and thus depends on access
control service and other security services.

1.5 SECURITY MECHANISMS

Table 1.3 lists the security mechanisms defined in X.800. The mechanisms are
divided into those that are implemented in a specific protocol layer, such as TCP or
an application-layer protocol, and those that are not specific to any particular pro-
tocol layer or security service. These mechanisms will be covered in the appropriate

SPECIFIC SECURITY MECHANISMS
May be incorporated into the appropriate protocol
layer in order to provide some of the OSI security
services.

Encipherment
The use of mathematical algorithms to transform
data into a form that is not readily intelligible. The
transformation and subsequent recovery of the data
depend on an algorithm and zero or more encryption
keys.

Digital Signature
Data appended to, or a cryptographic transformation
of, a data unit that allows a recipient of the data unit
to prove the source and integrity of the data unit and
protect against forgery (e.g., by the recipient).

Access Control
A variety of mechanisms that enforce access rights to
resources.

Data Integrity
A variety of mechanisms used to assure the integrity
of a data unit or stream of data units.

PERVASIVE SECURITY MECHANISMS

Mechanisms that are not specific to any particular
OSI security service or protocol layer.

Trusted Functionality
That which is perceived to be correct with respect
to some criteria (e.g., as established by a security
policy).

Security Label
The marking bound to a resource (which may be a
data unit) that names or designates the security attri-
butes of that resource.

Event Detection
Detection of security-relevant events.

Security Audit Trail
Data collected and potentially used to facilitate a
security audit, which is an independent review and
examination of system records and activities.

Security Recovery
Deals with requests from mechanisms, such as event
handling and management functions, and takes
recovery actions.

Table 1.3 Security Mechanisms (X.800)

1.5 / SECURITY MECHANISMS 33

places in the book. So we do not elaborate now, except to comment on the defini-
tion of encipherment. X.800 distinguishes between reversible encipherment mech-
anisms and irreversible encipherment mechanisms. A reversible encipherment
mechanism is simply an encryption algorithm that allows data to be encrypted and
subsequently decrypted. Irreversible encipherment mechanisms include hash algo-
rithms and message authentication codes, which are used in digital signature and
message authentication applications.

Table 1.4, based on one in X.800, indicates the relationship between security
services and security mechanisms.

SPECIFIC SECURITY MECHANISMS

Authentication Exchange
A mechanism intended to ensure the identity of an
entity by means of information exchange.

Traffic Padding
The insertion of bits into gaps in a data stream to
frustrate traffic analysis attempts.

Routing Control
Enables selection of particular physically secure
routes for certain data and allows routing changes,
especially when a breach of security is suspected.

Notarization
The use of a trusted third party to assure certain
properties of a data exchange.

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Peer entity authentication

SERVICE

MECHANISM

En
cip

he
rm

en
t

Di
git

al
sig

na
tur

e

Ac
ces

s c
on

tro
l

Da
ta

int
eg

rit
y

Au
the

nti
cat

ion
ex

ch
an

ge

Tr
affi

c p
ad

din
g

Ro
uti

ng
co

ntr
ol

No
tar

iza
tio

n

Data origin authentication

Access control

Confidentiality

Traffic flow confidentiality

Data integrity

Nonrepudiation

Availability

Table 1.4 Relationship Between Security Services and Mechanisms

34 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

1.6 FUNDAMENTAL SECURITY DESIGN PRINCIPLES

Despite years of research and development, it has not been possible to develop
security design and implementation techniques that systematically exclude security
flaws and prevent all unauthorized actions. In the absence of such foolproof tech-
niques, it is useful to have a set of widely agreed design principles that can guide
the development of protection mechanisms. The National Centers of Academic
Excellence in Information Assurance/Cyber Defense, which is jointly sponsored by
the U.S. National Security Agency and the U.S. Department of Homeland Security,
list the following as fundamental security design principles [NCAE13]:

■ Economy of mechanism

■ Fail-safe defaults

■ Complete mediation

■ Open design

■ Separation of privilege

■ Least privilege

■ Least common mechanism

■ Psychological acceptability

■ Isolation

■ Encapsulation

■ Modularity

■ Layering

■ Least astonishment

The first eight listed principles were first proposed in [SALT75] and have withstood
the test of time. In this section, we briefly discuss each principle.

Economy of mechanism means that the design of security measures embod-
ied in both hardware and software should be as simple and small as possible.
The motivation for this principle is that relatively simple, small design is eas-
ier to test and verify thoroughly. With a complex design, there are many more
opportunities for an adversary to discover subtle weaknesses to exploit that may
be difficult to spot ahead of time. The more complex the mechanism, the more
likely it is to possess exploitable flaws. Simple mechanisms tend to have fewer
exploitable flaws and require less maintenance. Further, because configuration
management issues are simplified, updating or replacing a simple mechanism
becomes a less intensive process. In practice, this is perhaps the most difficult
principle to honor. There is a constant demand for new features in both hard-
ware and software, complicating the security design task. The best that can be
done is to keep this principle in mind during system design to try to eliminate
unnecessary complexity.

Fail-safe defaults means that access decisions should be based on permission
rather than exclusion. That is, the default situation is lack of access, and the protec-
tion scheme identifies conditions under which access is permitted. This approach

1.6 / FUNDAMENTAL SECURITY DESIGN PRINCIPLES 35

exhibits a better failure mode than the alternative approach, where the default is
to permit access. A design or implementation mistake in a mechanism that gives
explicit permission tends to fail by refusing permission, a safe situation that can
be quickly detected. On the other hand, a design or implementation mistake in a
mechanism that explicitly excludes access tends to fail by allowing access, a failure
that may long go unnoticed in normal use. Most file access systems and virtually all
protected services on client/server systems use fail-safe defaults.

Complete mediation means that every access must be checked against the
access control mechanism. Systems should not rely on access decisions retrieved
from a cache. In a system designed to operate continuously, this principle requires
that, if access decisions are remembered for future use, careful consideration be
given to how changes in authority are propagated into such local memories. File
access systems appear to provide an example of a system that complies with this
principle. However, typically, once a user has opened a file, no check is made to see
if permissions change. To fully implement complete mediation, every time a user
reads a field or record in a file, or a data item in a database, the system must exercise
access control. This resource-intensive approach is rarely used.

Open design means that the design of a security mechanism should be open
rather than secret. For example, although encryption keys must be secret, encryption
algorithms should be open to public scrutiny. The algorithms can then be reviewed
by many experts, and users can therefore have high confidence in them. This is the
philosophy behind the National Institute of Standards and Technology (NIST)
program of standardizing encryption and hash algorithms, and has led to the wide-
spread adoption of NIST-approved algorithms.

Separation of privilege is defined in [SALT75] as a practice in which mul-
tiple privilege attributes are required to achieve access to a restricted resource.
A good example of this is multifactor user authentication, which requires the use of
multiple techniques, such as a password and a smart card, to authorize a user. The
term is also now applied to any technique in which a program is divided into parts
that are limited to the specific privileges they require in order to perform a specific
task. This is used to mitigate the potential damage of a computer security attack.
One example of this latter interpretation of the principle is removing high privilege
operations to another process and running that process with the higher privileges
required to perform its tasks. Day-to-day interfaces are executed in a lower privi-
leged process.

Least privilege means that every process and every user of the system should
operate using the least set of privileges necessary to perform the task. A good
example of the use of this principle is role-based access control. The system security
policy can identify and define the various roles of users or processes. Each role is
assigned only those permissions needed to perform its functions. Each permission
specifies a permitted access to a particular resource (such as read and write access
to a specified file or directory, connect access to a given host and port). Unless a
permission is granted explicitly, the user or process should not be able to access the
protected resource. More generally, any access control system should allow each
user only the privileges that are authorized for that user. There is also a temporal
aspect to the least privilege principle. For example, system programs or administra-
tors who have special privileges should have those privileges only when necessary;

36 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

when they are doing ordinary activities the privileges should be withdrawn. Leaving
them in place just opens the door to accidents.

Least common mechanism means that the design should minimize the func-
tions shared by different users, providing mutual security. This principle helps
reduce the number of unintended communication paths and reduces the amount of
hardware and software on which all users depend, thus making it easier to verify if
there are any undesirable security implications.

Psychological acceptability implies that the security mechanisms should not
interfere unduly with the work of users, while at the same time meeting the needs of
those who authorize access. If security mechanisms hinder the usability or accessibil-
ity of resources, then users may opt to turn off those mechanisms. Where possible,
security mechanisms should be transparent to the users of the system or at most
introduce minimal obstruction. In addition to not being intrusive or burdensome,
security procedures must reflect the user’s mental model of protection. If the protec-
tion procedures do not make sense to the user or if the user must translate his image
of protection into a substantially different protocol, the user is likely to make errors.

Isolation is a principle that applies in three contexts. First, public access sys-
tems should be isolated from critical resources (data, processes, etc.) to prevent dis-
closure or tampering. In cases where the sensitivity or criticality of the information
is high, organizations may want to limit the number of systems on which that data is
stored and isolate them, either physically or logically. Physical isolation may include
ensuring that no physical connection exists between an organization’s public access
information resources and an organization’s critical information. When implement-
ing logical isolation solutions, layers of security services and mechanisms should be
established between public systems and secure systems responsible for protecting
critical resources. Second, the processes and files of individual users should be iso-
lated from one another except where it is explicitly desired. All modern operating
systems provide facilities for such isolation, so that individual users have separate,
isolated process space, memory space, and file space, with protections for prevent-
ing unauthorized access. And finally, security mechanisms should be isolated in the
sense of preventing access to those mechanisms. For example, logical access control
may provide a means of isolating cryptographic software from other parts of the
host system and for protecting cryptographic software from tampering and the keys
from replacement or disclosure.

Encapsulation can be viewed as a specific form of isolation based on object-
oriented functionality. Protection is provided by encapsulating a collection of pro-
cedures and data objects in a domain of its own so that the internal structure of a
data object is accessible only to the procedures of the protected subsystem, and the
procedures may be called only at designated domain entry points.

Modularity in the context of security refers both to the development of security
functions as separate, protected modules and to the use of a modular architecture for
mechanism design and implementation. With respect to the use of separate security
modules, the design goal here is to provide common security functions and services,
such as cryptographic functions, as common modules. For example, numerous proto-
cols and applications make use of cryptographic functions. Rather than implement-
ing such functions in each protocol or application, a more secure design is provided
by developing a common cryptographic module that can be invoked by numerous

1.7 / ATTACK SURFACES AND ATTACK TREES 37

protocols and applications. The design and implementation effort can then focus on
the secure design and implementation of a single cryptographic module and includ-
ing mechanisms to protect the module from tampering. With respect to the use of a
modular architecture, each security mechanism should be able to support migration
to new technology or upgrade of new features without requiring an entire system
redesign. The security design should be modular so that individual parts of the secu-
rity design can be upgraded without the requirement to modify the entire system.

Layering refers to the use of multiple, overlapping protection approaches
addressing the people, technology, and operational aspects of information systems.
By using multiple, overlapping protection approaches, the failure or circumven-
tion of any individual protection approach will not leave the system unprotected.
We will see throughout this book that a layering approach is often used to provide
multiple barriers between an adversary and protected information or services. This
technique is often referred to as defense in depth.

Least astonishment means that a program or user interface should always
respond in the way that is least likely to astonish the user. For example, the mechanism
for authorization should be transparent enough to a user that the user has a good intui-
tive understanding of how the security goals map to the provided security mechanism.

1.7 ATTACK SURFACES AND ATTACK TREES

In Section 1.3, we provided an overview of the spectrum of security threats and
attacks facing computer and network systems. Section 22.1 goes into more detail
about the nature of attacks and the types of adversaries that present security threats.
In this section, we elaborate on two concepts that are useful in evaluating and clas-
sifying threats: attack surfaces and attack trees.

Attack Surfaces

An attack surface consists of the reachable and exploitable vulnerabilities in a sys-
tem [MANA11, HOWA03]. Examples of attack surfaces are the following:

■ Open ports on outward facing Web and other servers, and code listening on
those ports

■ Services available on the inside of a firewall

■ Code that processes incoming data, email, XML, office documents, and indus-
try-specific custom data exchange formats

■ Interfaces, SQL, and Web forms

■ An employee with access to sensitive information vulnerable to a social
engineering attack

Attack surfaces can be categorized as follows:

■ Network attack surface: This category refers to vulnerabilities over an enterprise
network, wide-area network, or the Internet. Included in this category are net-
work protocol vulnerabilities, such as those used for a denial-of-service attack,
disruption of communications links, and various forms of intruder attacks.

Hiva-Network.Com

38 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

■ Software attack surface: This refers to vulnerabilities in application, utility,
or operating system code. A particular focus in this category is Web server
software.

■ Human attack surface: This category refers to vulnerabilities created by
personnel or outsiders, such as social engineering, human error, and trusted
insiders.

An attack surface analysis is a useful technique for assessing the scale and
severity of threats to a system. A systematic analysis of points of vulnerability
makes developers and security analysts aware of where security mechanisms are
required. Once an attack surface is defined, designers may be able to find ways to
make the surface smaller, thus making the task of the adversary more difficult. The
attack surface also provides guidance on setting priorities for testing, strengthening
security measures, and modifying the service or application.

As illustrated in Figure 1.3, the use of layering, or defense in depth, and attack
surface reduction complement each other in mitigating security risk.

Attack Trees

An attack tree is a branching, hierarchical data structure that represents a set of poten-
tial techniques for exploiting security vulnerabilities [MAUW05, MOOR01, SCHN99].
The security incident that is the goal of the attack is represented as the root node of
the tree, and the ways that an attacker could reach that goal are iteratively and incre-
mentally represented as branches and subnodes of the tree. Each subnode defines a
subgoal, and each subgoal may have its own set of further subgoals, and so on. The
final nodes on the paths outward from the root, that is, the leaf nodes, represent differ-
ent ways to initiate an attack. Each node other than a leaf is either an AND-node or an
OR-node. To achieve the goal represented by an AND-node, the subgoals represented
by all of that node’s subnodes must be achieved; and for an OR-node, at least one of
the subgoals must be achieved. Branches can be labeled with values representing dif-
ficulty, cost, or other attack attributes, so that alternative attacks can be compared.

Figure 1.3 Defense in Depth and Attack Surface

Attack surface

Medium
security risk

High
security risk

Low
security riskD

ee
p

L
ay

er
in

g

Sh
al

lo
w

Small Large

Medium
security risk

1.7 / ATTACK SURFACES AND ATTACK TREES 39

The motivation for the use of attack trees is to effectively exploit the infor-
mation available on attack patterns. Organizations such as CERT publish security
advisories that have enabled the development of a body of knowledge about both
general attack strategies and specific attack patterns. Security analysts can use the
attack tree to document security attacks in a structured form that reveals key vul-
nerabilities. The attack tree can guide both the design of systems and applications,
and the choice and strength of countermeasures.

Figure 1.4, based on a figure in [DIMI07], is an example of an attack tree
analysis for an Internet banking authentication application. The root of the tree is
the objective of the attacker, which is to compromise a user’s account. The shaded
boxes on the tree are the leaf nodes, which represent events that comprise the
attacks. Note that in this tree, all the nodes other than leaf nodes are OR-nodes.
The analysis to generate this tree considered the three components involved in
authentication:

Figure 1.4 An Attack Tree for Internet Banking Authentication

Bank account compromise

User credential compromise

User credential guessing

UT/U1a User surveillance

UT/U1b Theft of token and
handwritten notes

Malicious software
installation Vulnerability exploit

UT/U2a Hidden code

UT/U2b Worms

UT/U3a Smartcard analyzers

UT/U2c Emails with
malicious code

UT/U3b Smartcard reader
manipulator

UT/U3c Brute force attacks
with PIN calculators

CC2 Sniffing

UT/U4a Social engineering

IBS3 Web site manipulation

UT/U4b Web page
obfuscation

CC1 Pharming

Redirection of
communication toward
fraudulent site

CC3 Active man-in-the
middle attacks

IBS1 Brute force attacks

User communication
with attacker

Injection of commands

Use of known authenticated
session by attacker

Normal user authentication
with specified session ID

CC4 Pre-defined session
IDs (session hijacking)

IBS2 Security policy
violation

40 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

■ User terminal and user (UT/U): These attacks target the user equipment,
including the tokens that may be involved, such as smartcards or other pass-
word generators, as well as the actions of the user.

■ Communications channel (CC): This type of attack focuses on communica-
tion links.

■ Internet banking server (IBS): These types of attacks are offline attacks against
the servers that host the Internet banking application.

Five overall attack strategies can be identified, each of which exploits one or
more of the three components. The five strategies are as follows:

■ User credential compromise: This strategy can be used against many ele-
ments of the attack surface. There are procedural attacks, such as monitoring
a user’s action to observe a PIN or other credential, or theft of the user’s
token or handwritten notes. An adversary may also compromise token
information using a variety of token attack tools, such as hacking the smart-
card or using a brute force approach to guess the PIN. Another possible
strategy is to embed malicious software to compromise the user’s login and
password. An adversary may also attempt to obtain credential information
via the communication channel (sniffing). Finally, an adversary may use
various means to engage in communication with the target user, as shown
in Figure 1.4.

■ Injection of commands: In this type of attack, the attacker is able to intercept
communication between the UT and the IBS. Various schemes can be used
to be able to impersonate the valid user and so gain access to the banking
system.

■ User credential guessing: It is reported in [HILT06] that brute force attacks
against some banking authentication schemes are feasible by sending ran-
dom usernames and passwords. The attack mechanism is based on distributed
zombie personal computers, hosting automated programs for username- or
password-based calculation.

■ Security policy violation: For example, violating the bank’s security policy
in combination with weak access control and logging mechanisms, an em-
ployee may cause an internal security incident and expose a customer’s
account.

■ Use of known authenticated session: This type of attack persuades or forces
the user to connect to the IBS with a preset session ID. Once the user authen-
ticates to the server, the attacker may utilize the known session ID to send
packets to the IBS, spoofing the user’s identity.

Figure 1.4 provides a thorough view of the different types of attacks on an
Internet banking authentication application. Using this tree as a starting point, secu-
rity analysts can assess the risk of each attack and, using the design principles out-
lined in the preceding section, design a comprehensive security facility. [DIMO07]
provides a good account of the results of this design effort.

1.8 / A MODEL FOR NETWORK SECURITY 41

1.8 A MODEL FOR NETWORK SECURITY

A model for much of what we will be discussing is captured, in very general terms, in
Figure 1.5. A message is to be transferred from one party to another across some sort
of Internet service. The two parties, who are the principals in this transaction, must
cooperate for the exchange to take place. A logical information channel is established
by defining a route through the Internet from source to destination and by the coop-
erative use of communication protocols (e.g., TCP/IP) by the two principals.

Security aspects come into play when it is necessary or desirable to protect the
information transmission from an opponent who may present a threat to confidentiality,
authenticity, and so on. All the techniques for providing security have two components:

■ A security-related transformation on the information to be sent. Examples
include the encryption of the message, which scrambles the message so that it
is unreadable by the opponent, and the addition of a code based on the con-
tents of the message, which can be used to verify the identity of the sender.

■ Some secret information shared by the two principals and, it is hoped,
unknown to the opponent. An example is an encryption key used in conjunc-
tion with the transformation to scramble the message before transmission
and unscramble it on reception.6

A trusted third party may be needed to achieve secure transmission. For
example, a third party may be responsible for distributing the secret information

6Part Two discusses a form of encryption, known as a symmetric encryption, in which only one of the two
principals needs to have the secret information.

Figure 1.5 Model for Network Security

Information
channelSecurity-related

transformation

Sender

Secret
information

M
es

sa
ge

M
es

sa
ge

Se
cu

re
m

es
sa

ge

Se
cu

re
m

es
sa

ge

Recipient

Opponent

Trusted third party
(e.g., arbiter, distributer
of secret information)

Security-related
transformation

Secret
information

42 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

to the two principals while keeping it from any opponent. Or a third party may be
needed to arbitrate disputes between the two principals concerning the authenticity
of a message transmission.

This general model shows that there are four basic tasks in designing a par-
ticular security service:

1. Design an algorithm for performing the security-related transformation. The
algorithm should be such that an opponent cannot defeat its purpose.

2. Generate the secret information to be used with the algorithm.

3. Develop methods for the distribution and sharing of the secret information.

4. Specify a protocol to be used by the two principals that makes use of the
security algorithm and the secret information to achieve a particular security
service.

Parts One through Five of this book concentrate on the types of security
mechanisms and services that fit into the model shown in Figure 1.5. However,
there are other security-related situations of interest that do not neatly fit this
model but are considered in this book. A general model of these other situations
is illustrated in Figure 1.6, which reflects a concern for protecting an information
system from unwanted access. Most readers are familiar with the concerns caused
by the existence of hackers, who attempt to penetrate systems that can be accessed
over a network. The hacker can be someone who, with no malign intent, simply gets
satisfaction from breaking and entering a computer system. The intruder can be a
disgruntled employee who wishes to do damage or a criminal who seeks to exploit
computer assets for financial gain (e.g., obtaining credit card numbers or perform-
ing illegal money transfers).

Another type of unwanted access is the placement in a computer system of
logic that exploits vulnerabilities in the system and that can affect application pro-
grams as well as utility programs, such as editors and compilers. Programs can pres-
ent two kinds of threats:

■ Information access threats: Intercept or modify data on behalf of users who
should not have access to that data.

■ Service threats: Exploit service flaws in computers to inhibit use by legitimate
users.

Figure 1.6 Network Access Security Model

Computing resources
(processor, memory, I/O)

Data

Processes

Software

Internal security controls

Information system

Gatekeeper
function

Opponent
—human (e.g., hacker)
—software
(e.g., virus, worm)

Access channel

1.9 / STANDARDS 43

Viruses and worms are two examples of software attacks. Such attacks can be
introduced into a system by means of a disk that contains the unwanted logic con-
cealed in otherwise useful software. They can also be inserted into a system across a
network; this latter mechanism is of more concern in network security.

The security mechanisms needed to cope with unwanted access fall into two
broad categories (see Figure 1.6). The first category might be termed a gatekeeper
function. It includes password-based login procedures that are designed to deny
access to all but authorized users and screening logic that is designed to detect and
reject worms, viruses, and other similar attacks. Once either an unwanted user
or unwanted software gains access, the second line of defense consists of a vari-
ety of internal controls that monitor activity and analyze stored information in an
attempt to detect the presence of unwanted intruders. These issues are explored
in Part Six.

1.9 STANDARDS

Many of the security techniques and applications described in this book have been
specified as standards. Additionally, standards have been developed to cover man-
agement practices and the overall architecture of security mechanisms and services.
Throughout this book, we describe the most important standards in use or that are
being developed for various aspects of cryptography and network security. Various
organizations have been involved in the development or promotion of these stan-
dards. The most important (in the current context) of these organizations are as
follows:

■ National Institute of Standards and Technology: NIST is a U.S. federal agency
that deals with measurement science, standards, and technology related to
U.S. government use and to the promotion of U.S. private-sector innovation.
Despite its national scope, NIST Federal Information Processing Standards
(FIPS) and Special Publications (SP) have a worldwide impact.

■ Internet Society: ISOC is a professional membership society with world-
wide organizational and individual membership. It provides leadership in
addressing issues that confront the future of the Internet and is the organiza-
tion home for the groups responsible for Internet infrastructure standards,
including the Internet Engineering Task Force (IETF) and the Internet
Architecture Board (IAB). These organizations develop Internet stan-
dards and related specifications, all of which are published as Requests for
Comments (RFCs).

■ ITU-T: The International Telecommunication Union (ITU) is an interna-
tional organization within the United Nations System in which governments
and the private sector coordinate global telecom networks and services. The
ITU Telecommunication Standardization Sector (ITU-T) is one of the three
sectors of the ITU. ITU-T’s mission is the development of technical standards
covering all fields of telecommunications. ITU-T standards are referred to as
Recommendations.

44 CHAPTER 1 / COMPUTER AND NETWORK SECURITY CONCEPTS

■ ISO: The International Organization for Standardization (ISO)7 is a world-
wide federation of national standards bodies from more than 140 countries,
one from each country. ISO is a nongovernmental organization that promotes
the development of standardization and related activities with a view to fa-
cilitating the international exchange of goods and services and to developing
cooperation in the spheres of intellectual, scientific, technological, and eco-
nomic activity. ISO’s work results in international agreements that are pub-
lished as International Standards.

A more detailed discussion of these organizations is contained in Appendix D.

1.10 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

7ISO is not an acronym (in which case it would be IOS), but it is a word, derived from the Greek, mean-
ing equal.

Key Terms

access control
active attack
authentication
authenticity
availability
data confidentiality
data integrity

denial of service
encryption
integrity
intruder
masquerade
nonrepudiation
OSI security architecture

passive attack
replay
security attacks
security mechanisms
security services
traffic analysis

Review Questions

1.1 What is the OSI security architecture?
1.2 List and briefly define the three key objectives of computer security.
1.3 List and briefly define categories of passive and active security attacks.
1.4 List and briefly define categories of security services.
1.5 List and briefly define categories of security mechanisms.
1.6 List and briefly define the fundamental security design principles.
1.7 Explain the difference between an attack surface and an attack tree.

Problems

1.1 Consider an automated cash deposit machine in which users provide a card or an ac-
count number to deposit cash. Give examples of confidentiality, integrity, and avail-
ability requirements associated with the system, and, in each case, indicate the degree
of importance of the requirement.

1.2 Repeat Problem 1.1 for a payment gateway system where a user pays for an item
using their account via the payment gateway.

1.10 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 45

1.3 Consider a financial report publishing system used to produce reports for various
organizations.
a. Give an example of a type of publication in which confidentiality of the stored

data is the most important requirement.
b. Give an example of a type of publication in which data integrity is the most im-

portant requirement.
c. Give an example in which system availability is the most important requirement.

1.4 For each of the following assets, assign a low, moderate, or high impact level for the
loss of confidentiality, availability, and integrity, respectively. Justify your answers.
a. A student maintaining a blog to post public information.
b. An examination section of a university that is managing sensitive information

about exam papers.
c. An information system in a pathological laboratory maintaining the patient’s data.
d. A student information system used for maintaining student data in a university

that contains both personal, academic information and routine administrative in-
formation (not privacy related). Assess the impact for the two data sets separately
and the information system as a whole.

e. A University library contains a library management system which controls the
distribution of books amongst the students of various departments. The library
management system contains both the student data and the book data. Assess the
impact for the two data sets separately and the information system as a whole.

1.5 Draw a matrix similar to Table 1.4 that shows the relationship between security ser-
vices and attacks.

1.6 Draw a matrix similar to Table 1.4 that shows the relationship between security
mechanisms and attacks.

1.7 Develop an attack tree for gaining access to the contents of a physical safe.
1.8 Consider a company whose operations are housed in two buildings on the same prop-

erty; one building is headquarters, the other building contains network and computer
services. The property is physically protected by a fence around the perimeter, and
the only entrance to the property is through this fenced perimeter. In addition to
the perimeter fence, physical security consists of a guarded front gate. The local net-
works are split between the Headquarters’ LAN and the Network Services’ LAN.
Internet users connect to the Web server through a firewall. Dial-up users get access
to a particular server on the Network Services’ LAN. Develop an attack tree in which
the root node represents disclosure of proprietary secrets. Include physical, social
engineering, and technical attacks. The tree may contain both AND and OR nodes.
Develop a tree that has at least 15 leaf nodes.

1.9 Read all of the classic papers cited in the Recommended Reading section for this
chapter, available at the Author Web site at WilliamStallings.com/Cryptography. The
papers are available at box.com/Crypto7e. Compose a 500–1000 word paper (or 8–12
slide PowerPoint presentation) that summarizes the key concepts that emerge from
these papers, emphasizing concepts that are common to most or all of the papers.

4646

2.1 Divisibility and The Division Algorithm
Divisibility
The Division Algorithm

2.2 The Euclidean Algorithm
Greatest Common Divisor
Finding the Greatest Common Divisor

2.3 Modular Arithmetic
The Modulus
Properties of Congruences
Modular Arithmetic Operations
Properties of Modular Arithmetic
Euclidean Algorithm Revisited
The Extended Euclidean Algorithm

2.4 Prime Numbers

2.5 Fermat’s and Euler’s Theorems

Fermat’s Theorem
Euler’s Totient Function
Euler’s Theorem

2.6 Testing for Primality

Miller–Rabin Algorithm
A Deterministic Primality Algorithm
Distribution of Primes

2.7 The Chinese Remainder Theorem

2.8 Discrete Logarithms

The Powers of an Integer, Modulo n
Logarithms for Modular Arithmetic
Calculation of Discrete Logarithms

2.9 Key Terms, Review Questions, and Problems

Appendix 2A The Meaning of Mod

CHAPTER

Introduction to Number Theory

Hiva-Network.Com

2.1 / DIVISIBILITY AND THE DIVISION ALGORITHM 47

Number theory is pervasive in cryptographic algorithms. This chapter provides
sufficient breadth and depth of coverage of relevant number theory topics for under-
standing the wide range of applications in cryptography. The reader familiar with these
topics can safely skip this chapter.

The first three sections introduce basic concepts from number theory that are
needed for understanding finite fields; these include divisibility, the Euclidian algo-
rithm, and modular arithmetic. The reader may study these sections now or wait until
ready to tackle Chapter 5 on finite fields.

Sections 2.4 through 2.8 discuss aspects of number theory related to prime num-
bers and discrete logarithms. These topics are fundamental to the design of asymmetric
(public-key) cryptographic algorithms. The reader may study these sections now or
wait until ready to read Part Three.

The concepts and techniques of number theory are quite abstract, and it is often
difficult to grasp them intuitively without examples. Accordingly, this chapter includes
a number of examples, each of which is highlighted in a shaded box.

2.1 DIVISIBILITY AND THE DIVISION ALGORITHM

Divisibility

We say that a nonzero b divides a if a = mb for some m, where a, b, and m are
integers. That is, b divides a if there is no remainder on division. The notation b � a
is commonly used to mean b divides a. Also, if b � a, we say that b is a divisor of a.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Understand the concept of divisibility and the division algorithm.

◆ Understand how to use the Euclidean algorithm to find the greatest com-
mon divisor.

◆ Present an overview of the concepts of modular arithmetic.

◆ Explain the operation of the extended Euclidean algorithm.

◆ Discuss key concepts relating to prime numbers.

◆ Understand Fermat’s theorem.

◆ Understand Euler’s theorem.

◆ Define Euler’s totient function.

◆ Make a presentation on the topic of testing for primality.

◆ Explain the Chinese remainder theorem.

◆ Define discrete logarithms.

48 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

Subsequently, we will need some simple properties of divisibility for integers,
which are as follows:

■ If a � 1, then a = {1.
■ If a �b and b � a, then a = {b.
■ Any b ≠ 0 divides 0.
■ If a �b and b � c, then a � c:

The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

13 � 182; -5 � 30; 17 � 289; -3 � 33; 17 � 0

11 � 66 and 66 � 198 1 11 � 198

b = 7; g = 14; h = 63; m = 3; n = 2
7 � 14 and 7 � 63.
To show 7 � (3 * 14 + 2 * 63),
we have (3 * 14 + 2 * 63) = 7(3 * 2 + 2 * 9),
and it is obvious that 7 � (7(3 * 2 + 2 * 9)).

■ If b � g and b �h, then b � (mg + nh) for arbitrary integers m and n.

To see this last point, note that

■ If b � g, then g is of the form g = b * g1 for some integer g1.
■ If b �h, then h is of the form h = b * h1 for some integer h1.

So

mg + nh = mbg1 + nbh1 = b * (mg1 + nh1)

and therefore b divides mg + nh.

The Division Algorithm

Given any positive integer n and any nonnegative integer a, if we divide a by n,
we get an integer quotient q and an integer remainder r that obey the following
relationship:

a = qn + r 0 … r 6 n; q = :a/n; (2.1)
where :x; is the largest integer less than or equal to x. Equation (2.1) is referred to
as the division algorithm.1

1Equation (2.1) expresses a theorem rather than an algorithm, but by tradition, this is referred to as the
division algorithm.

2.2 / THE EUCLIDEAN ALGORITHM 49

Figure 2.1a demonstrates that, given a and positive n, it is always possible to
find q and r that satisfy the preceding relationship. Represent the integers on the
number line; a will fall somewhere on that line (positive a is shown, a similar dem-
onstration can be made for negative a). Starting at 0, proceed to n, 2n, up to qn, such
that qn … a and (q + 1)n 7 a. The distance from qn to a is r, and we have found
the unique values of q and r. The remainder r is often referred to as a residue.

a = 11; n = 7; 11 = 1 * 7 + 4; r = 4 q = 1
a = -11; n = 7; -11 = (-2) * 7 + 3; r = 3 q = -2

Figure 2.1b provides another example.

Figure 2.1 The Relationship a = qn + r; 0 … r 6 n

0

n 2n 3n qn (q + 1)na

n

r(a) General relationship

0 15

15

10

30
= 2 × 15

70

(b) Example: 70 = (4 × 15) + 10

45
= 3 × 15

60
= 4 × 15

75
= 5 × 15

2.2 THE EUCLIDEAN ALGORITHM

One of the basic techniques of number theory is the Euclidean algorithm, which
is a simple procedure for determining the greatest common divisor of two positive
integers. First, we need a simple definition: Two integers are relatively prime if and
only if their only common positive integer factor is 1.

Greatest Common Divisor

Recall that nonzero b is defined to be a divisor of a if a = mb for some m, where
a, b, and m are integers. We will use the notation gcd(a, b) to mean the greatest
common divisor of a and b. The greatest common divisor of a and b is the largest
integer that divides both a and b. We also define gcd(0, 0) = 0.

50 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

More formally, the positive integer c is said to be the greatest common divisor
of a and b if

1. c is a divisor of a and of b.

2. any divisor of a and b is a divisor of c.

An equivalent definition is the following:

gcd(a, b) = max[k, such that k � a and k �b]

Because we require that the greatest common divisor be positive, gcd(a, b) =
gcd(a, -b) = gcd(-a, b) = gcd(-a, -b). In general, gcd(a, b) = gcd( � a � , �b � ).

gcd(60, 24) = gcd(60, -24) = 12

8 and 15 are relatively prime because the positive divisors of 8 are 1, 2, 4, and 8, and
the positive divisors of 15 are 1, 3, 5, and 15. So 1 is the only integer on both lists.

Also, because all nonzero integers divide 0, we have gcd(a, 0) = � a � .
We stated that two integers a and b are relatively prime if and only if their

only common positive integer factor is 1. This is equivalent to saying that a and b are
relatively prime if gcd(a, b) = 1.

Finding the Greatest Common Divisor

We now describe an algorithm credited to Euclid for easily finding the greatest
common divisor of two integers (Figure 2.2). This algorithm has broad significance
in cryptography. The explanation of the algorithm can be broken down into the fol-
lowing points:

1. Suppose we wish to determine the greatest common divisor d of the integers
a and b; that is determine d = gcd(a, b). Because gcd( � a � , �b � ) = gcd(a, b),
there is no harm in assuming a Ú b 7 0.

2. Dividing a by b and applying the division algorithm, we can state:

a = q1b + r1 0 … r1 6 b (2.2)

3. First consider the case in which r1 = 0. Therefore b divides a and clearly no
larger number divides both b and a, because that number would be larger
than b. So we have d = gcd(a, b) = b.

4. The other possibility from Equation (2.2) is r1 ≠ 0. For this case, we can state
that d � r1. This is due to the basic properties of divisibility: the relations d � a
and d �b together imply that d � (a - q1b), which is the same as d � r1.

5. Before proceeding with the Euclidian algorithm, we need to answer the ques-
tion: What is the gcd(b, r1)? We know that d �b and d � r1. Now take any arbi-
trary integer c that divides both b and r1. Therefore, c � (q1b + r1) = a. Because
c divides both a and b, we must have c … d, which is the greatest common
divisor of a and b. Therefore d = gcd(b, r1).

2.2 / THE EUCLIDEAN ALGORITHM 51

Let us now return to Equation (2.2) and assume that r1 ≠ 0. Because b 7 r1,
we can divide b by r1 and apply the division algorithm to obtain:

b = q2r1 + r2 0 … r2 6 r1

As before, if r2 = 0, then d = r1 and if r2 ≠ 0, then d = gcd(r1, r2). Note that the
remainders form a descending series of nonnegative values and so must terminate
when the remainder is zero. This happens, say, at the (n + 1)th stage where rn - 1 is
divided by rn. The result is the following system of equations:

a = q1b + r1 0 6 r1 6 b
b = q2r1 + r2 0 6 r2 6 r1
r1 = q3r2 + r3 0 6 r3 6 r2

~ ~

~ ~
~ ~

rn - 2 = qnrn - 1 + rn 0 6 rn 6 rn - 1
rn - 1 = qn + 1rn + 0
d = gcd(a, b) = rn

w (2.3)
At each iteration, we have d = gcd(ri, ri+ 1) until finally d = gcd(rn, 0) = rn.

Thus, we can find the greatest common divisor of two integers by repetitive appli-
cation of the division algorithm. This scheme is known as the Euclidean algorithm.
Figure 2.3 illustrates a simple example.

We have essentially argued from the top down that the final result is the
gcd(a, b). We can also argue from the bottom up. The first step is to show that rn
divides a and b. It follows from the last division in Equation (2.3) that rn divides
rn - 1. The next to last division shows that rn divides rn - 2 because it divides both

Figure 2.2 Euclidean Algorithm

No

No Yes
a > b?

r > 0?
Swap

a and b

Replace
b with r

Replace
a with b

Divide a by b,
calling the

remainder r

GCD is
the final

value of b

START

END Figure 2.3 Euclidean
Algorithm Example:
gcd(710, 310)

710 = 2 × 310 + 90

310 = 3 × 90 + 40

90 = 2 × 40 + 10

40 = 4 × 10

GCDGCD

Same GCD

52 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

terms on the right. Successively, one sees that rn divides all ri>s and finally a and b.
It remains to show that rn is the largest divisor that divides a and b. If we take any
arbitrary integer that divides a and b, it must also divide r1, as explained previously.
We can follow the sequence of equations in Equation (2.3) down and show that c
must divide all ri>s. Therefore c must divide rn, so that rn = gcd(a, b).

Let us now look at an example with relatively large numbers to see the power
of this algorithm:

To find d = gcd(a, b) = gcd(1160718174, 316258250)

a = q1b + r1 1160718174 = 3 * 316258250 + 211943424 d = gcd(316258250, 211943424)
b = q2r1 + r2 316258250 = 1 * 211943424 + 104314826 d = gcd(211943424, 104314826)
r1 = q3r2 + r3 211943424 = 2 * 104314826 + 3313772 d = gcd(104314826, 3313772)
r2 = q4r3 + r4 104314826 = 31 * 3313772 + 1587894 d = gcd(3313772, 1587894)
r3 = q5r4 + r5 3313772 = 2 * 1587894 + 137984 d = gcd(1587894, 137984)
r4 = q6r5 + r6 1587894 = 11 * 137984 + 70070 d = gcd(137984, 70070)
r5 = q7r6 + r7 137984 = 1 * 70070 + 67914 d = gcd(70070, 67914)
r6 = q8r7 + r8 70070 = 1 * 67914 + 2156 d = gcd(67914, 2156)
r7 = q9r8 + r9 67914 = 31 * 2156 + 1078 d = gcd(2156, 1078)
r8 = q10r9 + r10 2156 = 2 * 1078 + 0 d = gcd(1078, 0) = 1078
Therefore, d = gcd(1160718174, 316258250) = 1078

In this example, we begin by dividing 1160718174 by 316258250, which gives 3
with a remainder of 211943424. Next we take 316258250 and divide it by 211943424.
The process continues until we get a remainder of 0, yielding a result of 1078.

It will be helpful in what follows to recast the above computation in tabular
form. For every step of the iteration, we have ri- 2 = qiri- 1 + ri, where ri- 2 is the
dividend, ri- 1 is the divisor, qi is the quotient, and ri is the remainder. Table 2.1 sum-
marizes the results.

Dividend Divisor Quotient Remainder

a = 1160718174 b = 316258250 q1 = 3 r1 = 211943424

b = 316258250 r1 = 211943434 q2 = 1 r2 = 104314826

r1 = 211943424 r2 = 104314826 q3 = 2 r3 = 3313772

r2 = 104314826 r3 = 3313772 q4 = 31 r4 = 1587894

r3 = 3313772 r4 = 1587894 q5 = 2 r5 = 137984

r4 = 1587894 r5 = 137984 q6 = 11 r6 = 70070

r5 = 137984 r6 = 70070 q7 = 1 r7 = 67914

r6 = 70070 r7 = 67914 q8 = 1 r8 = 2156

r7 = 67914 r8 = 2156 q9 = 31 r9 = 1078

r8 = 2156 r9 = 1078 q10 = 2 r10 = 0

Table 2.1 Euclidean Algorithm Example

2.3 / MODULAR ARITHMETIC 53

2.3 MODULAR ARITHMETIC

The Modulus

If a is an integer and n is a positive integer, we define a mod n to be the remainder
when a is divided by n. The integer n is called the modulus. Thus, for any integer a,
we can rewrite Equation (2.1) as follows:

a = qn + r 0 … r 6 n; q = :a/n;
a = :a/n; * n + (a mod n)

11 mod 7 = 4; -11 mod 7 = 3

73 K 4 (mod 23); 21 K -9 (mod 10)

Two integers a and b are said to be congruent modulo n, if (a mod n) =
(b mod n). This is written as a K b (mod n).2

2We have just used the operator mod in two different ways: first as a binary operator that produces a re-
mainder, as in the expression a mod b; second as a congruence relation that shows the equivalence of two
integers, as in the expression a K b (mod n). See Appendix 2A for a discussion.

Note that if a K 0 (mod n), then n � a.

Properties of Congruences

Congruences have the following properties:

1. a K b (mod n) if n � (a - b).
2. a K b (mod n) implies b K a (mod n).
3. a K b (mod n) and b K c (mod n) imply a K c (mod n).

To demonstrate the first point, if n � (a - b), then (a - b) = kn for some k.
So we can write a = b + kn. Therefore, (a mod n) = (remainder when b +
kn is divided by n) = (remainder when b is divided by n) = (b mod n).

23 K 8 (mod 5) because 23 - 8 = 15 = 5 * 3
-11 K 5 (mod 8) because -11 - 5 = -16 = 8 * (-2)
81 K 0 (mod 27) because 81 - 0 = 81 = 27 * 3

The remaining points are as easily proved.

54 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

Modular Arithmetic Operations

Note that, by definition (Figure 2.1), the (mod n) operator maps all integers into
the set of integers {0, 1, c , (n - 1)}. This suggests the question: Can we perform
arithmetic operations within the confines of this set? It turns out that we can; this
technique is known as modular arithmetic.

Modular arithmetic exhibits the following properties:

1. [(a mod n) + (b mod n)] mod n = (a + b) mod n
2. [(a mod n) - (b mod n)] mod n = (a - b) mod n
3. [(a mod n) * (b mod n)] mod n = (a * b) mod n

We demonstrate the first property. Define (a mod n) = ra and (b mod n) = rb.
Then we can write a = ra + jn for some integer j and b = rb + kn for some integer k.
Then

(a + b) mod n = (ra + jn + rb + kn) mod n
= (ra + rb + (k + j)n) mod n
= (ra + rb) mod n
= [(a mod n) + (b mod n)] mod n

The remaining properties are proven as easily. Here are examples of the three
properties:

11 mod 8 = 3; 15 mod 8 = 7
[(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2
(11 + 15) mod 8 = 26 mod 8 = 2
[(11 mod 8) - (15 mod 8)] mod 8 = -4 mod 8 = 4
(11 - 15) mod 8 = -4 mod 8 = 4
[(11 mod 8) * (15 mod 8)] mod 8 = 21 mod 8 = 5
(11 * 15) mod 8 = 165 mod 8 = 5

To find 117 mod 13, we can proceed as follows:

112 = 121 K 4 (mod 13)
114 = (112)2 K 42 K 3 (mod 13)
117 = 11 * 112 * 114

117 K 11 * 4 * 3 K 132 K 2 (mod 13)

Exponentiation is performed by repeated multiplication, as in ordinary
arithmetic.

Thus, the rules for ordinary arithmetic involving addition, subtraction, and
multiplication carry over into modular arithmetic.

2.3 / MODULAR ARITHMETIC 55

Table 2.2 provides an illustration of modular addition and multiplication
modulo 8. Looking at addition, the results are straightforward, and there is a reg-
ular pattern to the matrix. Both matrices are symmetric about the main diagonal
in conformance to the commutative property of addition and multiplication. As in
ordinary addition, there is an additive inverse, or negative, to each integer in modu-
lar arithmetic. In this case, the negative of an integer x is the integer y such that
(x + y) mod 8 = 0. To find the additive inverse of an integer in the left-hand col-
umn, scan across the corresponding row of the matrix to find the value 0; the integer
at the top of that column is the additive inverse; thus, (2 + 6) mod 8 = 0. Similarly,
the entries in the multiplication table are straightforward. In modular arithmetic mod
8, the multiplicative inverse of x is the integer y such that (x * y) mod 8 = 1 mod 8.
Now, to find the multiplicative inverse of an integer from the multiplication table,
scan across the matrix in the row for that integer to find the value 1; the integer at
the top of that column is the multiplicative inverse; thus, (3 * 3) mod 8 = 1. Note
that not all integers mod 8 have a multiplicative inverse; more about that later.

Properties of Modular Arithmetic

Define the set Zn as the set of nonnegative integers less than n:

Zn = {0, 1, c , (n - 1)}

Table 2.2 Arithmetic Modulo 8
+ 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 7

1 1 2 3 4 5 6 7 0

2 2 3 4 5 6 7 0 1

3 3 4 5 6 7 0 1 2

4 4 5 6 7 0 1 2 3

5 5 6 7 0 1 2 3 4

6 6 7 0 1 2 3 4 5

7 7 0 1 2 3 4 5 6

(a) Addition modulo 8

* 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7

2 0 2 4 6 0 2 4 6

3 0 3 6 1 4 7 2 5

4 0 4 0 4 0 4 0 4

5 0 5 2 7 4 1 6 3

6 0 6 4 2 0 6 4 2

7 0 7 6 5 4 3 2 1

(b) Multiplication modulo 8

w -w w-1

0 0 —

1 7 1

2 6 —

3 5 3

4 4 —

5 3 5

6 2 —

7 1 7

(c) Additive and multiplicative
inverse modulo 8

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56 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

This is referred to as the set of residues, or residue classes (mod n). To be more pre-
cise, each integer in Zn represents a residue class. We can label the residue classes
(mod n) as [0], [1], [2], c , [n - 1], where

[r] = {a: a is an integer, a K r (mod n)}

The residue classes (mod 4) are

[0] = {c , -16, -12, -8, -4, 0, 4, 8, 12, 16, c }
[1] = {c , -15, -11, -7, -3, 1, 5, 9, 13, 17, c }
[2] = {c , -14, -10, -6, -2, 2, 6, 10, 14, 18, c }
[3] = {c , -13, -9, -5, -1, 3, 7, 11, 15, 19, c }

Property Expression

Commutative Laws
(w + x) mod n = (x + w) mod n
(w * x) mod n = (x * w) mod n

Associative Laws
[(w + x) + y] mod n = [w + (x + y)] mod n
[(w * x) * y] mod n = [w * (x * y)] mod n

Distributive Law [w * (x + y)] mod n = [(w * x) + (w * y)] mod n

Identities
(0 + w) mod n = w mod n
(1 * w) mod n = w mod n

Additive Inverse (-w) For each w∈ Zn, there exists a z such that w + z K 0 mod n

Table 2.3 Properties of Modular Arithmetic for Integers in Zn

Of all the integers in a residue class, the smallest nonnegative integer is the
one used to represent the residue class. Finding the smallest nonnegative integer to
which k is congruent modulo n is called reducing k modulo n.

If we perform modular arithmetic within Zn, the properties shown in Table 2.3
hold for integers in Zn. We show in the next section that this implies that Zn is a
commutative ring with a multiplicative identity element.

There is one peculiarity of modular arithmetic that sets it apart from ordinary
arithmetic. First, observe that (as in ordinary arithmetic) we can write the following:

if (a + b) K (a + c) (mod n) then b K c (mod n) (2.4)

(5 + 23) K (5 + 7)(mod 8); 23 K 7(mod 8)

Equation (2.4) is consistent with the existence of an additive inverse. Adding
the additive inverse of a to both sides of Equation (2.4), we have

((-a) + a + b) K ((-a) + a + c)(mod n)
b K c (mod n)

2.3 / MODULAR ARITHMETIC 57

However, the following statement is true only with the attached condition:

if (a * b) K (a * c)(mod n) then b K c(mod n) if a is relatively prime to n (2.5)

Recall that two integers are relatively prime if their only common positive integer
factor is 1. Similar to the case of Equation (2.4), we can say that Equation (2.5) is
consistent with the existence of a multiplicative inverse. Applying the multiplicative
inverse of a to both sides of Equation (2.5), we have

((a-1)ab) K ((a-1)ac)(mod n)
b K c(mod n)

To see this, consider an example in which the condition of Equation (2.5) does not
hold. The integers 6 and 8 are not relatively prime, since they have the common
factor 2. We have the following:

6 * 3 = 18 K 2(mod 8)
6 * 7 = 42 K 2(mod 8)

Yet 3 [ 7 (mod 8).

The reason for this strange result is that for any general modulus n, a multi-
plier a that is applied in turn to the integers 0 through (n - 1) will fail to produce a
complete set of residues if a and n have any factors in common.

With a = 6 and n = 8,

Z8 0 1 2 3 4 5 6 7
Multiply by 6 0 6 12 18 24 30 36 42
Residues 0 6 4 2 0 6 4 2

Because we do not have a complete set of residues when multiplying by
6, more than one integer in Z8 maps into the same residue. Specifically,
6 * 0 mod 8 = 6 * 4 mod 8; 6 * 1 mod 8 = 6 * 5 mod 8; and so on. Because
this is a many-to-one mapping, there is not a unique inverse to the multiply
operation.

However, if we take a = 5 and n = 8, whose only common factor is 1,

Z8 0 1 2 3 4 5 6 7
Multiply by 5 0 5 10 15 20 25 30 35
Residues 0 5 2 7 4 1 6 3

The line of residues contains all the integers in Z8, in a different order.

58 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

In general, an integer has a multiplicative inverse in Zn if and only if that inte-
ger is relatively prime to n. Table 2.2c shows that the integers 1, 3, 5, and 7 have a
multiplicative inverse in Z8; but 2, 4, and 6 do not.

Euclidean Algorithm Revisited

The Euclidean algorithm can be based on the following theorem: For any integers
a, b, with a Ú b Ú 0,

gcd(a, b) = gcd(b, a mod b) (2.6)

gcd(55, 22) = gcd(22, 55 mod 22) = gcd(22, 11) = 11

gcd(18, 12) = gcd(12, 6) = gcd(6, 0) = 6
gcd(11, 10) = gcd(10, 1) = gcd(1, 0) = 1

To see that Equation (2.6) works, let d = gcd(a, b). Then, by the definition of
gcd, d � a and d �b. For any positive integer b, we can express a as

a = kb + r K r (mod b)
a mod b = r

with k, r integers. Therefore, (a mod b) = a - kb for some integer k. But because
d �b, it also divides kb. We also have d � a. Therefore, d � (a mod b). This shows that
d is a common divisor of b and (a mod b). Conversely, if d is a common divisor of b
and (a mod b), then d �kb and thus d � [kb + (a mod b)], which is equivalent to d � a.
Thus, the set of common divisors of a and b is equal to the set of common divisors
of b and (a mod b). Therefore, the gcd of one pair is the same as the gcd of the other
pair, proving the theorem.

Equation (2.6) can be used repetitively to determine the greatest common divisor.

This is the same scheme shown in Equation (2.3), which can be rewritten in
the following way.

Euclidean Algorithm

Calculate Which satisfies

r1 = a mod b a = q1b + r1
r2 = b mod r1 b = q2r1 + r2
r3 = r1 mod r2 r1 = q3r2 + r3

~

~

~

~

~

~

rn = rn - 2 mod rn - 1 rn - 2 = qnrn - 1 + rn
rn + 1 = rn - 1 mod rn = 0 rn - 1 = qn + 1rn + 0

d = gcd(a, b) = rn

We can define the Euclidean algorithm concisely as the following recursive
function.

2.3 / MODULAR ARITHMETIC 59

Euclid(a,b)
if (b=0) then return a;
else return Euclid(b, a mod b);

The Extended Euclidean Algorithm

We now proceed to look at an extension to the Euclidean algorithm that will be
important for later computations in the area of finite fields and in encryption algo-
rithms, such as RSA. For given integers a and b, the extended Euclidean algorithm
not only calculates the greatest common divisor d but also two additional integers x
and y that satisfy the following equation.

ax + by = d = gcd(a, b) (2.7)

It should be clear that x and y will have opposite signs. Before examining the
algorithm, let us look at some of the values of x and y when a = 42 and b = 30.
Note that gcd(42, 30) = 6. Here is a partial table of values3 for 42x + 30y.

x − 3 − 2 − 1 0 1 2 3

y

-3 -216 -174 -132 -90 -48 -6 36
-2 -186 -144 -102 -60 -18 24 66
-1 -156 -114 -72 -30 12 54 96

0 -126 -84 -42 0 42 84 126
1 -96 -54 -12 30 72 114 156
2 -66 -24 18 60 102 144 186
3 -36 6 48 90 132 174 216

Observe that all of the entries are divisible by 6. This is not surpris-
ing, because both 42 and 30 are divisible by 6, so every number of the form
42x + 30y = 6(7x + 5y) is a multiple of 6. Note also that gcd(42, 30) = 6 appears
in the table. In general, it can be shown that for given integers a and b, the smallest
positive value of ax + by is equal to gcd(a, b).

Now let us show how to extend the Euclidean algorithm to determine (x, y, d)
given a and b. We again go through the sequence of divisions indicated in Equation
(2.3), and we assume that at each step i we can find integers xi and yi that satisfy
ri = axi + byi. We end up with the following sequence.

a = q1b + r1 r1 = ax1 + by1
b = q2r1 + r2 r2 = ax2 + by2
r1 = q3r2 + r3 r3 = ax3 + by3

f f
rn - 2 = qnrn - 1 + rn rn = axn + byn
rn - 1 = qn + 1rn + 0

3This example is taken from [SILV06].

60 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

Now, observe that we can rearrange terms to write

ri = ri- 2 - ri- 1qi (2.8)

Also, in rows i - 1 and i - 2, we find the values

ri- 2 = axi- 2 + byi- 2 and ri- 1 = axi- 1 + byi- 1

Substituting into Equation (2.8), we have

ri = (axi- 2 + byi- 2) - (axi- 1 + byi- 1)qi
= a(xi- 2 - qixi- 1) + b(yi- 2 - qiyi- 1)

But we have already assumed that ri = axi + byi. Therefore,

xi = xi- 2 - qixi- 1 and yi = yi- 2 - qiyi- 1

We now summarize the calculations:

Extended Euclidean Algorithm

Calculate Which satisfies Calculate Which satisfies

r-1 = a x-1 = 1; y-1 = 0 a = ax-1 + by-1
r0 = b x0 = 0; y0 = 1 b = ax0 + by0
r1 = a mod b
q1 = :a/b;

a = q1b + r1 x1 = x-1 - q1x0 = 1
y1 = y-1 - q1y0 = -q1

r1 = ax1 + by1

r2 = b mod r1
q2 = :b/r1;

b = q2r1 + r2 x2 = x0 - q2x1
y2 = y0 - q2y1

r2 = ax2 + by2

r3 = r1 mod r2
q3 = :r1/r2;

r1 = q3r2 + r3 x3 = x1 - q3x2
y3 = y1 - q3y2

r3 = ax3 + by3

~

~

~

~

~

~

~

~

~

~

~

~

rn = rn - 2 mod rn - 1
qn = :rn - 2/rn - 1;

rn - 2 = qnrn - 1 + rn xn = xn - 2 - qnxn - 1
yn = yn - 2 - qnyn - 1

rn = axn + byn

rn + 1 = rn - 1 mod rn = 0
qn + 1 = :rn - 1/rn;

rn - 1 = qn + 1rn + 0 d = gcd(a, b) = rn
x = xn; y = yn

We need to make several additional comments here. In each row, we calculate
a new remainder ri based on the remainders of the previous two rows, namely ri- 1
and ri- 2. To start the algorithm, we need values for r0 and r-1, which are just a and b.
It is then straightforward to determine the required values for x-1, y-1, x0, and y0.

We know from the original Euclidean algorithm that the process ends
with a remainder of zero and that the greatest common divisor of a and b is
d = gcd(a, b) = rn. But we also have determined that d = rn = axn + byn.
Therefore, in Equation (2.7), x = xn and y = yn.

As an example, let us use a = 1759 and b = 550 and solve for
1759x + 550y = gcd(1759, 550). The results are shown in Table 2.4. Thus, we have
1759 * (-111) + 550 * 355 = -195249 + 195250 = 1.

2.4 / PRIME NUMBERS 61

2.4 PRIME NUMBERS4

A central concern of number theory is the study of prime numbers. Indeed, whole
books have been written on the subject (e.g., [CRAN01], [RIBE96]). In this section,
we provide an overview relevant to the concerns of this book.

An integer p 7 1 is a prime number if and only if its only divisors5 are {1 and
{p. Prime numbers play a critical role in number theory and in the techniques dis-
cussed in this chapter. Table 2.5 shows the primes less than 2000. Note the way the
primes are distributed. In particular, note the number of primes in each range of
100 numbers.

Any integer a 7 1 can be factored in a unique way as

a = p1a1 * p2a2 * g * ptat (2.9)

where p1 6 p2 6 c 6 pt are prime numbers and where each ai is a positive inte-
ger. This is known as the fundamental theorem of arithmetic; a proof can be found
in any text on number theory.

4In this section, unless otherwise noted, we deal only with the nonnegative integers. The use of negative
integers would introduce no essential differences.
5Recall from Section 2.1 that integer a is said to be a divisor of integer b if there is no remainder on
division. Equivalently, we say that a divides b.

i ri qi xi yi

-1 1759 1 0

0 550 0 1

1 109 3 1 -3

2 5 5 -5 16

3 4 21 106 -339

4 1 1 -111 355

5 0 4

Result: d = 1; x = -111; y = 355

Table 2.4 Extended Euclidean Algorithm Example

91 = 7 * 13
3600 = 24 * 32 * 52

11011 = 7 * 112 * 13

It is useful for what follows to express this another way. If P is the set of
all prime numbers, then any positive integer a can be written uniquely in the
following form:

a = q
p∈P

pap where each ap Ú 0

62 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

2
10

1
21

1
30

7
40

1
50

3
60

1
70

1
80

9
90

7
10

09
11

03
12

01
13

01
14

09
15

11
16

01
17

09
18

01
19

01

3
10

3
22

3
31

1
40

9
50

9
60

7
70

9
81

1
91

1
10

13
11

09
12

13
13

03
14

23
15

23
16

07
17

21
18

11
19

07

5
10

7
22

7
31

3
41

9
52

1
61

3
71

9
82

1
91

9
10

19
11

17
12

17
13

07
14

27
15

31
16

09
17

23
18

23
19

13

7
10

9
22

9
31

7
42

1
52

3
61

7
72

7
82

3
92

9
10

21
11

23
12

23
13

19
14

29
15

43
16

13
17

33
18

31
19

31

11
11

3
23

3
33

1
43

1
54

1
61

9
73

3
82

7
93

7
10

31
11

29
12

29
13

21
14

33
15

49
16

19
17

41
18

47
19

33

13
12

7
23

9
33

7
43

3
54

7
63

1
73

9
82

9
94

1
10

33
11

51
12

31
13

27
14

39
15

53
16

21
17

47
18

61
19

49

17
13

1
24

1
34

7
43

9
55

7
64

1
74

3
83

9
94

7
10

39
11

53
12

37
13

61
14

47
15

59
16

27
17

53
18

67
19

51

19
13

7
25

1
34

9
44

3
56

3
64

3
75

1
85

3
95

3
10

49
11

63
12

49
13

67
14

51
15

67
16

37
17

59
18

71
19

73

23
13

9
25

7
35

3
44

9
56

9
64

7
75

7
85

7
96

7
10

51
11

71
12

59
13

73
14

53
15

71
16

57
17

77
18

73
19

79

29
14

9
26

3
35

9
45

7
57

1
65

3
76

1
85

9
97

1
10

61
11

81
12

77
13

81
14

59
15

79
16

63
17

83
18

77
19

87

31
15

1
26

9
36

7
46

1
57

7
65

9
76

9
86

3
97

7
10

63
11

87
12

79
13

99
14

71
15

83
16

67
17

87
18

79
19

93

37
15

7
27

1
37

3
46

3
58

7
66

1
77

3
87

7
98

3
10

69
11

93
12

83
14

81
15

97
16

69
17

89
18

89
19

97

41
16

3
27

7
37

9
46

7
59

3
67

3
78

7
88

1
99

1
10

87
12

89
14

83
16

93
19

99

43
16

7
28

1
38

3
47

9
59

9
67

7
79

7
88

3
99

7
10

91
12

91
14

87
16

97

47
17

3
28

3
38

9
48

7
68

3
88

7
10

93
12

97
14

89
16

99

53
17

9
29

3
39

7
49

1
69

1
10

97
14

93

59
18

1
49

9
14

99

61
19

1

67
19

3

71
19

7

73
19

9

79 83 89 97

T
ab

le
2

.5

P
ri

m
es

U
nd

er
2

00
0

2.4 / PRIME NUMBERS 63

The right-hand side is the product over all possible prime numbers p; for any par-
ticular value of a, most of the exponents ap will be 0.

The value of any given positive integer can be specified by simply listing all the
nonzero exponents in the foregoing formulation.

The integer 12 is represented by {a2 = 2, a3 = 1}.
The integer 18 is represented by {a2 = 1, a3 = 2}.
The integer 91 is represented by {a7 = 1, a13 = 1}.

Multiplication of two numbers is equivalent to adding the corresponding

exponents. Given a = q
p∈P

pap, b = q
p∈P

pbp. Define k = ab. We know that the inte-

ger k can be expressed as the product of powers of primes: k = q
p∈P

pkp. It follows
that kp = ap + bp for all p ∈ P.

k = 12 * 18 = (22 * 3) * (2 * 32) = 216
k2 = 2 + 1 = 3; k3 = 1 + 2 = 3
216 = 23 * 33 = 8 * 27

a = 12; b = 36; 12 � 36
12 = 22 * 3; 36 = 22 * 32

a2 = 2 = b2
a3 = 1 … 2 = b3
Thus, the inequality ap … bp is satisfied for all prime numbers.

What does it mean, in terms of the prime factors of a and b, to say that a divides b?
Any integer of the form pn can be divided only by an integer that is of a lesser
or equal power of the same prime number, pj with j … n. Thus, we can say the
following.

Given

a = q
p∈P

pap, b = q
p∈P

pbp

If a �b, then ap … bp for all p.

It is easy to determine the greatest common divisor of two positive integers if
we express each integer as the product of primes.

64 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

The following relationship always holds:

If k = gcd(a, b), then kp = min(ap, bp) for all p.

Determining the prime factors of a large number is no easy task, so the pre-
ceding relationship does not directly lead to a practical method of calculating the
greatest common divisor.

2.5 FERMAT’S AND EULER’S THEOREMS

Two theorems that play important roles in public-key cryptography are Fermat’s
theorem and Euler’s theorem.

Fermat’s Theorem6

Fermat’s theorem states the following: If p is prime and a is a positive integer not
divisible by p, then

ap - 1 K 1 (mod p) (2.10)

Proof: Consider the set of positive integers less than p: {1, 2, c , p - 1} and mul-
tiply each element by a, modulo p, to get the set X = {a mod p, 2a mod p, c ,
(p - 1)a mod p}. None of the elements of X is equal to zero because p does not
divide a. Furthermore, no two of the integers in X are equal. To see this, assume that
ja K ka(mod p)), where 1 … j 6 k … p - 1. Because a is relatively prime7 to p, we
can eliminate a from both sides of the equation [see Equation (2.3)] resulting in
j K k(mod p). This last equality is impossible, because j and k are both positive inte-
gers less than p. Therefore, we know that the (p - 1) elements of X are all positive
integers with no two elements equal. We can conclude the X consists of the set of
integers {1, 2, c , p - 1} in some order. Multiplying the numbers in both sets
(p and X) and taking the result mod p yields

a * 2a * g * (p - 1)a K [(1 * 2 * g * (p - 1)](mod p)
ap - 1(p - 1)! K (p - 1)! (mod p)

We can cancel the (p - 1)! term because it is relatively prime to p [see Equation
(2.5)]. This yields Equation (2.10), which completes the proof.

6This is sometimes referred to as Fermat’s little theorem.
7Recall from Section 2.2 that two numbers are relatively prime if they have no prime factors in common;
that is, their only common divisor is 1. This is equivalent to saying that two numbers are relatively prime
if their greatest common divisor is 1.

300 = 22 * 31 * 52

18 = 21 * 32

gcd(18,300) = 21 * 31 * 50 = 6

Hiva-Network.Com

2.5 / FERMAT’S AND EULER’S THEOREMS 65

An alternative form of Fermat’s theorem is also useful: If p is prime and a is a
positive integer, then

ap K a(mod p) (2.11)

Note that the first form of the theorem [Equation (2.10)] requires that a be rela-
tively prime to p, but this form does not.

a = 7, p = 19
72 = 49 K 11 (mod 19)
74 K 121 K 7 (mod 19)
78 K 49 K 11 (mod 19)
716 K 121 K 7 (mod 19)
ap - 1 = 718 = 716 * 72 K 7 * 11 K 1 (mod 19)

p = 5, a = 3 ap = 35 = 243 K 3(mod 5) = a(mod p)
p = 5, a = 10 ap = 105 = 100000 K 10(mod 5) K 0(mod 5) = a(mod p)

Euler’s Totient Function

Before presenting Euler’s theorem, we need to introduce an important quantity in
number theory, referred to as Euler’s totient function. This function, written f(n),
is defined as the number of positive integers less than n and relatively prime to n.
By convention, f(1) = 1.

Determine f(37) and f(35).

Because 37 is prime, all of the positive integers from 1 through 36 are relatively
prime to 37. Thus f(37) = 36.
To determine f(35), we list all of the positive integers less than 35 that are
relatively prime to it:

1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18

19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34

There are 24 numbers on the list, so f(35) = 24.

Table 2.6 lists the first 30 values of f(n). The value f(1) is without meaning
but is defined to have the value 1.

It should be clear that, for a prime number p,

f(p) = p - 1

Now suppose that we have two prime numbers p and q with p ≠ q. Then we can
show that, for n = pq,

66 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

f(n) = f(pq) = f(p) * f(q) = (p - 1) * (q - 1)

To see that f(n) = f(p) * f(q), consider that the set of positive integers less than
n is the set {1, c , (pq - 1)}. The integers in this set that are not relatively prime
to n are the set {p, 2p, c , (q - 1)p} and the set {q, 2q, c , (p - 1)q}. To see
this, consider that any integer that divides n must divide either of the prime num-
bers p or q. Therefore, any integer that does not contain either p or q as a factor is
relatively prime to n. Further note that the two sets just listed are non-overlapping:
Because p and q are prime, we can state that none of the integers in the first set can
be written as a multiple of q, and none of the integers in the second set can be writ-
ten as a multiple of p. Thus the total number of unique integers in the two sets is
(q - 1) + (p - 1). Accordingly,

f(n) = (pq - 1) - [(q - 1) + (p - 1)]
= pq - (p + q) + 1
= (p - 1) * (q - 1)
= f(p) * f(q)

f(21) = f(3) * f(7) = (3 - 1) * (7 - 1) = 2 * 6 = 12
where the 12 integers are {1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20}.

Table 2.6 Some Values of Euler’s Totient Function f(n)

n f(n)

1 1

2 1

3 2

4 2

5 4

6 2

7 6

8 4

9 6

10 4

n f(n)

11 10

12 4

13 12

14 6

15 8

16 8

17 16

18 6

19 18

20 8

n f(n)

21 12

22 10

23 22

24 8

25 20

26 12

27 18

28 12

29 28

30 8

Euler’s Theorem

Euler’s theorem states that for every a and n that are relatively prime:

af(n) K 1(mod n) (2.12)

Proof: Equation (2.12) is true if n is prime, because in that case, f(n) = (n - 1)
and Fermat’s theorem holds. However, it also holds for any integer n. Recall that

2.5 / FERMAT’S AND EULER’S THEOREMS 67

f(n) is the number of positive integers less than n that are relatively prime to n.
Consider the set of such integers, labeled as

R = {x1, x2, c , xf(n)}

That is, each element xi of R is a unique positive integer less than n with gcd(xi, n) = 1.
Now multiply each element by a, modulo n:

S = {(ax1 mod n), (ax2 mod n), c , (axf(n) mod n)}

The set S is a permutation8 of R , by the following line of reasoning:

1. Because a is relatively prime to n and xi is relatively prime to n, axi must also
be relatively prime to n. Thus, all the members of S are integers that are less
than n and that are relatively prime to n.

2. There are no duplicates in S. Refer to Equation (2.5). If axi mod n= axj
mod n, then xi = xj.

Therefore,

q
f(n)

i=1
(axi mod n) = q

f(n)

i=1
xi

q
f(n)

i=1
axi K q

f(n)

i=1
xi (mod n)

af(n) * Jqf(n)
i=1

xiR K qf(n)
i=1

xi (mod n)

af(n) K 1 (mod n)

which completes the proof. This is the same line of reasoning applied to the proof
of Fermat’s theorem.

8A permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each
element appearing exactly once.

a = 3; n = 10; f(10) = 4; af(n) = 34 = 81 = 1(mod 10) = 1(mod n)
a = 2; n = 11; f(11) = 10; af(n) = 210 = 1024 = 1(mod 11) = 1(mod n)

As is the case for Fermat’s theorem, an alternative form of the theorem is also
useful:

af(n) + 1 K a(mod n) (2.13)

Again, similar to the case with Fermat’s theorem, the first form of Euler’s theorem
[Equation (2.12)] requires that a be relatively prime to n, but this form does not.

68 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

2.6 TESTING FOR PRIMALITY

For many cryptographic algorithms, it is necessary to select one or more very large
prime numbers at random. Thus, we are faced with the task of determining whether
a given large number is prime. There is no simple yet efficient means of accomplish-
ing this task.

In this section, we present one attractive and popular algorithm. You may be
surprised to learn that this algorithm yields a number that is not necessarily a prime.
However, the algorithm can yield a number that is almost certainly a prime. This will
be explained presently. We also make reference to a deterministic algorithm for find-
ing primes. The section closes with a discussion concerning the distribution of primes.

Miller–Rabin Algorithm9

The algorithm due to Miller and Rabin [MILL75, RABI80] is typically used to test
a large number for primality. Before explaining the algorithm, we need some back-
ground. First, any positive odd integer n Ú 3 can be expressed as

n - 1 = 2kq with k 7 0, q odd

To see this, note that n - 1 is an even integer. Then, divide (n - 1) by 2 until the
result is an odd number q, for a total of k divisions. If n is expressed as a binary
number, then the result is achieved by shifting the number to the right until the
rightmost digit is a 1, for a total of k shifts. We now develop two properties of prime
numbers that we will need.

TWO PROPERTIES OF PRIME NUMBERS The first property is stated as follows: If p is
prime and a is a positive integer less than p, then a2 mod p = 1 if and only if either
a mod p = 1 or a mod p = -1 mod p = p - 1. By the rules of modular arithmetic
(a mod p) (a mod p) = a2 mod p. Thus, if either a mod p = 1 or a mod p = -1,
then a2 mod p = 1. Conversely, if a2 mod p = 1, then (a mod p)2 = 1, which is true
only for a mod p = 1 or a mod p = -1.

The second property is stated as follows: Let p be a prime number greater
than 2. We can then write p - 1 = 2kq with k 7 0, q odd. Let a be any integer in
the range 1 6 a 6 p - 1. Then one of the two following conditions is true.

1. aq is congruent to 1 modulo p. That is, aq mod p = 1, or equivalently,
aq K 1(mod p).

2. One of the numbers aq, a2q, a4q, c , a2
k - 1q is congruent to -1 mod-

ulo p. That is, there is some number j in the range (1 … j … k) such that
a2

j - 1q mod p = -1 mod p = p - 1 or equivalently, a2
j - 1q K - 1(mod p).

Proof: Fermat’s theorem [Equation (2.10)] states that an - 1 K 1(mod n) if n is
prime. We have p - 1 = 2kq. Thus, we know that ap - 1 mod p = a2

kq mod p = 1.
Thus, if we look at the sequence of numbers

aq mod p, a2q mod p, a4q mod p, c , a2
k - 1q mod p, a2

kq mod p (2.14)

9Also referred to in the literature as the Rabin-Miller algorithm, or the Rabin-Miller test, or the Miller–
Rabin test.

2.6 / TESTING FOR PRIMALITY 69

we know that the last number in the list has value 1. Further, each number in the list
is the square of the previous number. Therefore, one of the following possibilities
must be true.

1. The first number on the list, and therefore all subsequent numbers on the list,
equals 1.

2. Some number on the list does not equal 1, but its square mod p does equal 1.
By virtue of the first property of prime numbers defined above, we know that
the only number that satisfies this condition is p - 1. So, in this case, the list
contains an element equal to p - 1.

This completes the proof.

DETAILS OF THE ALGORITHM These considerations lead to the conclusion that,
if n is prime, then either the first element in the list of residues, or remainders,
(aq, a2q, c , a2

k - 1q, a2
kq) modulo n equals 1; or some element in the list equals

(n - 1); otherwise n is composite (i.e., not a prime). On the other hand, if the
condition is met, that does not necessarily mean that n is prime. For example, if
n = 2047 = 23 * 89, then n - 1 = 2 * 1023. We compute 21023 mod 2047 = 1, so
that 2047 meets the condition but is not prime.

We can use the preceding property to devise a test for primality. The procedure
TEST takes a candidate integer n as input and returns the result composite if n is
definitely not a prime, and the result inconclusive if n may or may not be a prime.

TEST (n)
1. Find integers k, q, with k > 0, q odd, so that

(n − 1 = 2k q);
2. Select a random integer a, 1 < a < n - 1;
3. if aq mod n = 1 then return(”inconclusive”);
4. for j = 0 to k - 1 do
5. if a2

j
qmod n = n - 1 then return(”inconclusive”);

6. return(”composite”);

Let us apply the test to the prime number n = 29. We have (n - 1) = 28 =
22(7) = 2kq. First, let us try a = 10. We compute 107 mod 29 = 17, which is neither
1 nor 28, so we continue the test. The next calculation finds that (107)2 mod 29 = 28,
and the test returns inconclusive (i.e., 29 may be prime). Let’s try again with
a = 2. We have the following calculations: 27 mod 29 = 12; 214 mod 29 = 28; and
the test again returns inconclusive. If we perform the test for all integers a in
the range 1 through 28, we get the same inconclusive result, which is compatible
with n being a prime number.

Now let us apply the test to the composite number n = 13 * 17 = 221. Then
(n - 1) = 220 = 22(55) = 2kq. Let us try a = 5. Then we have 555 mod 221 = 112,
which is neither 1 nor 220(555)2 mod 221 = 168. Because we have used all values of j
(i.e., j = 0 and j = 1) in line 4 of the TEST algorithm, the test returns composite, indi-
cating that 221 is definitely a composite number. But suppose we had selected a = 21.
Then we have 2155 mod 221 = 200; (2155)2 mod 221 = 220; and the test returns
inconclusive, indicating that 221 may be prime. In fact, of the 218 integers from 2
through 219, four of these will return an inconclusive result, namely 21, 47, 174, and 200.

70 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

REPEATED USE OF THE MILLER–RABIN ALGORITHM How can we use the Miller–Rabin
algorithm to determine with a high degree of confidence whether or not an integer
is prime? It can be shown [KNUT98] that given an odd number n that is not prime
and a randomly chosen integer, a with 1 6 a 6 n - 1, the probability that TEST
will return inconclusive (i.e., fail to detect that n is not prime) is less than 1/4.
Thus, if t different values of a are chosen, the probability that all of them will pass
TEST (return inconclusive) for n is less than (1/4)t. For example, for t = 10, the
probability that a nonprime number will pass all ten tests is less than 10-6. Thus,
for a sufficiently large value of t , we can be confident that n is prime if Miller’s test
always returns inconclusive.

This gives us a basis for determining whether an odd integer n is prime with
a reasonable degree of confidence. The procedure is as follows: Repeatedly invoke
TEST (n) using randomly chosen values for a. If, at any point, TEST returns
composite, then n is determined to be nonprime. If TEST continues to return
inconclusive for t tests, then for a sufficiently large value of t, assume that n
is prime.

A Deterministic Primality Algorithm

Prior to 2002, there was no known method of efficiently proving the primality of
very large numbers. All of the algorithms in use, including the most popular (Miller–
Rabin), produced a probabilistic result. In 2002 (announced in 2002, published
in 2004), Agrawal, Kayal, and Saxena [AGRA04] developed a relatively simple
deterministic algorithm that efficiently determines whether a given large number
is a prime. The algorithm, known as the AKS algorithm, does not appear to be as
efficient as the Miller–Rabin algorithm. Thus far, it has not supplanted this older,
probabilistic technique.

Distribution of Primes

It is worth noting how many numbers are likely to be rejected before a prime num-
ber is found using the Miller–Rabin test, or any other test for primality. A result
from number theory, known as the prime number theorem, states that the primes
near n are spaced on the average one every ln (n) integers. Thus, on average, one
would have to test on the order of ln(n) integers before a prime is found. Because
all even integers can be immediately rejected, the correct figure is 0.5 ln(n). For
example, if a prime on the order of magnitude of 2200 were sought, then about
0.5 ln(2200) = 69 trials would be needed to find a prime. However, this figure is just
an average. In some places along the number line, primes are closely packed, and in
other places there are large gaps.

The two consecutive odd integers 1,000,000,000,061 and 1,000,000,000,063
are both prime. On the other hand, 1001! + 2, 1001! + 3, c , 1001! + 1000,
1001! + 1001 is a sequence of 1000 consecutive composite integers.

2.7 / THE CHINESE REMAINDER THEOREM 71

2.7 THE CHINESE REMAINDER THEOREM

One of the most useful results of number theory is the Chinese remainder theorem
(CRT).10 In essence, the CRT says it is possible to reconstruct integers in a certain
range from their residues modulo a set of pairwise relatively prime moduli.

10The CRT is so called because it is believed to have been discovered by the Chinese mathematician
Sun-Tsu in around 100 A.D.

The 10 integers in Z10, that is the integers 0 through 9, can be reconstructed from
their two residues modulo 2 and 5 (the relatively prime factors of 10). Say the
known residues of a decimal digit x are r2 = 0 and r5 = 3; that is, x mod 2 = 0
and x mod 5 = 3. Therefore, x is an even integer in Z10 whose remainder, on divi-
sion by 5, is 3. The unique solution is x = 8.

The CRT can be stated in several ways. We present here a formulation that is most
useful from the point of view of this text. An alternative formulation is explored in
Problem 2.33. Let

M = q
k

i=1
mi

where the mi are pairwise relatively prime; that is, gcd(mi, mj) = 1 for 1 … i, j … k,
and i ≠ j. We can represent any integer A in ZM by a k-tuple whose elements are in
Zmi using the following correspondence:

A 4 (a1, a2, c , ak) (2.15)

where A ∈ ZM, ai∈ Zmi, and ai = A mod mi for 1 … i … k. The CRT makes two
assertions.

1. The mapping of Equation (2.15) is a one-to-one correspondence (called a
bijection) between ZM and the Cartesian product Zm1 * Zm2 * c * Zmk.
That is, for every integer A such that 0 … A 6 M, there is a unique k- tuple
(a1, a2, c , ak) with 0 … ai 6 mi that represents it, and for every such
k- tuple (a1, a2, c , ak), there is a unique integer A in ZM.

2. Operations performed on the elements of ZM can be equivalently performed
on the corresponding k-tuples by performing the operation independently in
each coordinate position in the appropriate system.

Let us demonstrate the first assertion. The transformation from A to
(a1, a2, c , ak), is obviously unique; that is, each ai is uniquely calculated as
ai = A mod mi. Computing A from (a1, a2, c , ak) can be done as follows. Let

72 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

Mi = M/mi for 1 … i … k. Note that Mi = m1 * m2 * c * mi- 1 * mi+ 1 * c
* mk, so that Mi K 0 (mod mj) for all j ≠ i. Then let

ci = Mi * (Mi-1 mod mi) for 1 … i … k (2.16)

By the definition of Mi, it is relatively prime to mi and therefore has a unique multi-
plicative inverse mod mi. So Equation (2.16) is well defined and produces a unique
value ci. We can now compute

A K ¢ ak
i=1

aici≤(mod M) (2.17)
To show that the value of A produced by Equation (2.17) is correct, we must

show that ai = A mod mi for 1 … i … k. Note that cj K Mj K 0 (mod mi) if j ≠ i,
and that ci K 1 (mod mi). It follows that ai = A mod mi.

The second assertion of the CRT, concerning arithmetic operations, follows
from the rules for modular arithmetic. That is, the second assertion can be stated as
follows: If

A 4 (a1, a2, c , ak)
B 4 (b1, b2, c , bk)

then

(A + B) mod M 4 ((a1 + b1) mod m1, c , (ak + bk) mod mk)
(A - B) mod M 4 ((a1 - b1) mod m1, c , (ak - bk) mod mk)
(A * B) mod M 4 ((a1 * b1) mod m1, c , (ak * bk) mod mk)

One of the useful features of the Chinese remainder theorem is that it provides
a way to manipulate (potentially very large) numbers mod M in terms of tuples of
smaller numbers. This can be useful when M is 150 digits or more. However, note
that it is necessary to know beforehand the factorization of M.

To represent 973 mod 1813 as a pair of numbers mod 37 and 49, define

m1 = 37
m2 = 49
M = 1813
A = 973

We also have M1 = 49 and M2 = 37. Using the extended Euclidean algorithm,
we compute M1

-1 = 34 mod m1 and M2-1 = 4 mod m2. (Note that we only need
to compute each Mi and each Mi

-1 once.) Taking residues modulo 37 and 49, our
representation of 973 is (11, 42), because 973 mod 37 = 11 and 973 mod 49 = 42.

Now suppose we want to add 678 to 973. What do we do to (11, 42)? First
we compute (678) 4 (678 mod 37, 678 mod 49) = (12, 41). Then we add the
tuples element-wise and reduce (11 + 12 mod 37, 42 + 41 mod 49) = (23, 34).
To verify that this has the correct effect, we compute

2.8 / DISCRETE LOGARITHMS 73

2.8 DISCRETE LOGARITHMS

Discrete logarithms are fundamental to a number of public-key algorithms, includ-
ing Diffie–Hellman key exchange and the digital signature algorithm (DSA). This
section provides a brief overview of discrete logarithms. For the interested reader,
more detailed developments of this topic can be found in [ORE67] and [LEVE90].

The Powers of an Integer, Modulo n

Recall from Euler’s theorem [Equation (2.12)] that, for every a and n that are rela-
tively prime,

af(n) K 1 (mod n)

where f(n), Euler’s totient function, is the number of positive integers less than n
and relatively prime to n. Now consider the more general expression:

am K 1 (mod n) (2.18)

If a and n are relatively prime, then there is at least one integer m that satisfies
Equation (2.18), namely, m = f(n). The least positive exponent m for which
Equation (2.18) holds is referred to in several ways:

■ The order of a (mod n)

■ The exponent to which a belongs (mod n)

■ The length of the period generated by a

(23, 34) 4 a1M1M1-1 + a2M2M2-1 mod M
= [(23)(49)(34) + (34)(37)(4)] mod 1813
= 43350 mod 1813
= 1651

and check that it is equal to (973 + 678) mod 1813 = 1651. Remember that in
the above derivation, Mi

-1 is the multiplicative inverse of M1 modulo m1 and M2
-1

is the multiplicative inverse of M2 modulo m2.
Suppose we want to multiply 1651 (mod 1813) by 73. We multiply (23, 34)

by 73 and reduce to get (23 * 73 mod 37, 34 * 73 mod 49) = (14, 32). It is eas-
ily verified that

(14, 32) 4 [(14)(49)(34) + (32)(37)(4)] mod 1813
= 865
= 1651 * 73 mod 1813

Hiva-Network.Com

74 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

Table 2.7 shows all the powers of a, modulo 19 for all positive a 6 19. The
length of the sequence for each base value is indicated by shading. Note the
following:

1. All sequences end in 1. This is consistent with the reasoning of the preceding
few paragraphs.

2. The length of a sequence divides f(19) = 18. That is, an integral number of
sequences occur in each row of the table.

3. Some of the sequences are of length 18. In this case, it is said that the base inte-
ger a generates (via powers) the set of nonzero integers modulo 19. Each such
integer is called a primitive root of the modulus 19.

More generally, we can say that the highest possible exponent to which a num-
ber can belong (mod n) is f(n). If a number is of this order, it is referred to as a
primitive root of n. The importance of this notion is that if a is a primitive root of n,
then its powers

a, a2, c , af(n)

are distinct (mod n) and are all relatively prime to n. In particular, for a prime num-
ber p, if a is a primitive root of p, then

a, a2, c , ap - 1

are distinct (mod p). For the prime number 19, its primitive roots are 2, 3, 10, 13, 14,
and 15.

Not all integers have primitive roots. In fact, the only integers with primitive
roots are those of the form 2, 4, pa, and 2pa, where p is any odd prime and a is a
positive integer. The proof is not simple but can be found in many number theory
books, including [ORE76].

To see this last point, consider the powers of 7, modulo 19:

71 K 7 (mod 19)
72 = 49 = 2 * 19 + 11 K 11 (mod 19)
73 = 343 = 18 * 19 + 1 K 1 (mod 19)
74 = 2401 = 126 * 19 + 7 K 7 (mod 19)
75 = 16807 = 884 * 19 + 11 K 11 (mod 19)

There is no point in continuing because the sequence is repeating. This can be
proven by noting that 73 K 1(mod 19), and therefore, 73 + j K 737j K 7j(mod 19),
and hence, any two powers of 7 whose exponents differ by 3 (or a multiple of 3)
are congruent to each other (mod 19). In other words, the sequence is periodic,
and the length of the period is the smallest positive exponent m such that
7m K 1(mod 19).

2.8 / DISCRETE LOGARITHMS 75

Logarithms for Modular Arithmetic

With ordinary positive real numbers, the logarithm function is the inverse of expo-
nentiation. An analogous function exists for modular arithmetic.

Let us briefly review the properties of ordinary logarithms. The logarithm of a
number is defined to be the power to which some positive base (except 1) must be
raised in order to equal the number. That is, for base x and for a value y,

y = xlogx(y)

The properties of logarithms include

logx(1) = 0
logx(x) = 1

logx(yz) = logx(y) + logx(z) (2.19)

logx(y
r) = r * logx(y) (2.20)

Consider a primitive root a for some prime number p (the argument can
be developed for nonprimes as well). Then we know that the powers of a from

a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 4 8 16 13 7 14 9 18 17 15 11 3 6 12 5 10 1

3 9 8 5 15 7 2 6 18 16 10 11 14 4 12 17 13 1

4 16 7 9 17 11 6 5 1 4 16 7 9 17 11 6 5 1

5 6 11 17 9 7 16 4 1 5 6 11 17 9 7 16 4 1

6 17 7 4 5 11 9 16 1 6 17 7 4 5 11 9 16 1

7 11 1 7 11 1 7 11 1 7 11 1 7 11 1 7 11 1

8 7 18 11 12 1 8 7 18 11 12 1 8 7 18 11 12 1

9 5 7 6 16 11 4 17 1 9 5 7 6 16 11 4 17 1

10 5 12 6 3 11 15 17 18 9 14 7 13 16 8 4 2 1

11 7 1 11 7 1 11 7 1 11 7 1 11 7 1 11 7 1

12 11 18 7 8 1 12 11 18 7 8 1 12 11 18 7 8 1

13 17 12 4 14 11 10 16 18 6 2 7 15 5 8 9 3 1

14 6 8 17 10 7 3 4 18 5 13 11 2 9 12 16 15 1

15 16 12 9 2 11 13 5 18 4 3 7 10 17 8 6 14 1

16 9 11 5 4 7 17 6 1 16 9 11 5 4 7 17 6 1

17 4 11 16 6 7 5 9 1 17 4 11 16 6 7 5 9 1

18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1

Table 2.7 Powers of Integers, Modulo 19

76 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

1 through (p - 1) produce each integer from 1 through (p - 1) exactly once. We
also know that any integer b satisfies

b K r (mod p) for some r, where 0 … r … (p - 1)

by the definition of modular arithmetic. It follows that for any integer b and a primi-
tive root a of prime number p, we can find a unique exponent i such that

b K ai(mod p) where 0 … i … (p - 1)

This exponent i is referred to as the discrete logarithm of the number b for the base
a (mod p). We denote this value as dloga,p(b).

11

Note the following:

dloga,p(1) = 0 because a0 mod p = 1 mod p = 1 (2.21)

dloga,p(a) = 1 because a1 mod p = a (2.22)

11Many texts refer to the discrete logarithm as the index. There is no generally agreed notation for this
concept, much less an agreed name.

Here is an example using a nonprime modulus, n = 9. Here f(n) = 6 and a = 2
is a primitive root. We compute the various powers of a and find

20 = 1 24 K 7 (mod 9)
21 = 2 25 K 5 (mod 9)
22 = 4 26 K 1 (mod 9)
23 = 8

This gives us the following table of the numbers with given discrete logarithms
(mod 9) for the root a = 2:

Logarithm 0 1 2 3 4 5
Number 1 2 4 8 7 5

To make it easy to obtain the discrete logarithms of a given number, we rearrange
the table:

Number 1 2 4 5 7 8
Logarithm 0 1 2 5 4 3

Now consider

x = adloga, p(x) mod p y = adloga, p(y) mod p
xy = adloga, p(xy) mod p

2.8 / DISCRETE LOGARITHMS 77

Using the rules of modular multiplication,

xy mod p = [(x mod p)(y mod p)] mod p

adloga, p(xy) mod p = [(adloga, p(x) mod p)(adloga, p(y) mod p)] mod p

= (adloga, p(x) + dloga, p(y)) mod p

But now consider Euler’s theorem, which states that, for every a and n that are
relatively prime,

af(n) K 1(mod n)

Any positive integer z can be expressed in the form z = q + kf(n), with
0 … q 6 f(n). Therefore, by Euler’s theorem,

az K aq(mod n) if z K q mod f(n)

Applying this to the foregoing equality, we have

dloga, p(xy) K [dloga, p(x) + dloga, p(y)](mod f(p))

and generalizing,

dloga, p(y
r) K [r * dloga, p(y)](mod f(p))

This demonstrates the analogy between true logarithms and discrete logarithms.
Keep in mind that unique discrete logarithms mod m to some base a exist only

if a is a primitive root of m.
Table 2.8, which is directly derived from Table 2.7, shows the sets of discrete

logarithms that can be defined for modulus 19.

Calculation of Discrete Logarithms

Consider the equation

y = gx mod p

Given g, x, and p, it is a straightforward matter to calculate y. At the worst, we must
perform x repeated multiplications, and algorithms exist for achieving greater effi-
ciency (see Chapter 9).

However, given y, g, and p, it is, in general, very difficult to calculate x (take
the discrete logarithm). The difficulty seems to be on the same order of magnitude
as that of factoring primes required for RSA. At the time of this writing, the asymp-
totically fastest known algorithm for taking discrete logarithms modulo a prime
number is on the order of [BETH91]:

e((ln p)
1/3(ln(ln p))2/3)

which is not feasible for large primes.

78 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

2.9 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

(a) Discrete logarithms to the base 2, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log2,19(a) 18 1 13 2 16 14 6 3 8 17 12 15 5 7 11 4 10 9

(b) Discrete logarithms to the base 3, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log3,19(a) 18 7 1 14 4 8 6 3 2 11 12 15 17 13 5 10 16 9

(c) Discrete logarithms to the base 10, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log10,19(a) 18 17 5 16 2 4 12 15 10 1 6 3 13 11 7 14 8 9

(d) Discrete logarithms to the base 13, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log13,19(a) 18 11 17 4 14 10 12 15 16 7 6 3 1 5 13 8 2 9

(e) Discrete logarithms to the base 14, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log14,19(a) 18 13 7 8 10 2 6 3 14 5 12 15 11 1 17 16 4 9

(f) Discrete logarithms to the base 15, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log15,19(a) 18 5 11 10 8 16 12 15 4 13 6 3 7 17 1 2 14 9

Table 2.8 Tables of Discrete Logarithms, Modulo 19

Key Terms

bijection
composite number
commutative
Chinese remainder theorem
discrete logarithm
divisor
Euclidean algorithm

Euler’s theorem
Euler’s totient function
Fermat’s theorem
greatest common divisor
identity element
index
modular arithmetic

modulus
order
prime number
primitive root
relatively prime
residue

2.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 79

Review Questions

2.1 What does it mean to say that b is a divisor of a?
2.2 What is the meaning of the expression a divides b?
2.3 What is the difference between modular arithmetic and ordinary arithmetic?
2.4 What is a prime number?
2.5 What is Euler’s totient function?
2.6 The Miller–Rabin test can determine if a number is not prime but cannot determine

if a number is prime. How can such an algorithm be used to test for primality?
2.7 What is a primitive root of a number?
2.8 What is the difference between an index and a discrete logarithm?

Problems

2.1 Reformulate Equation (2.1), removing the restriction that a is a nonnegative integer.
That is, let a be any integer.

2.2 Draw a figure similar to Figure 2.1 for a 6 0.
2.3 For each of the following equations, find an integer x that satisfies the equation.

a. 4 x K 2 (mod 3 )
b. 7 x K 4 (mod 9 )
c. 5 x K 3 (mod 1 1 )

2.4 In this text, we assume that the modulus is a positive integer. But the definition of the
expression a mod n also makes perfect sense if n is negative. Determine the following:
a. 7 mod 4
b. 7 mod -4
c. -7 mod 4
d. -7 mod -4

2.5 A modulus of 0 does not fit the definition but is defined by convention as follows:
a mod 0 = a. With this definition in mind, what does the following expression mean:
a K b (mod 0)?

2.6 In Section 2.3, we define the congruence relationship as follows: Two integers a and
b are said to be congruent modulo n if (a mod n) = (b mod n). We then proved that
a K b (mod n) if n � (a - b). Some texts on number theory use this latter relation-
ship as the definition of congruence: Two integers a and b are said to be congruent
modulo n if n � (a - b). Using this latter definition as the starting point, prove that, if
(a mod n) = (b mod n), then n divides (a - b).

2.7 What is the smallest positive integer that has exactly k divisors? Provide answers for
values for 1 … k … 8.

2.8 Prove the following:
a. a K b (mod n) implies b K a (mod n)
b. a K b (mod n) and b K c (mod n) imply a K c (mod n)

2.9 Prove the following:
a. [(a mod n) - (b mod n)] mod n = (a - b) mod n
b. [(a mod n) * (b mod n)] mod n = (a * b) mod n

2.10 Find the multiplicative inverse of each nonzero element in Z5.
2.11 Show that an integer N is congruent modulo 9 to the sum of its decimal digits. For

example, 7 2 3 K 7 + 2 + 3 K 1 2 K 1 + 2 K 3 (mod 9 ). This is the basis for the
familiar procedure of “casting out 9’s” when checking computations in arithmetic.

80 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

2.12 a. Determine gcd(72345, 43215)
b. Determine gcd(3486, 10292)

2.13 The purpose of this problem is to set an upper bound on the number of iterations of
the Euclidean algorithm.
a. Suppose that m = qn + r with q 7 0 and 0 … r 6 n. Show that m/2 7 r.
b. Let Ai be the value of A in the Euclidean algorithm after the ith iteration. Show that

Ai+ 2 6
Ai
2

c. Show that if m, n, and N are integers with (1 … m, n, … 2N), then the Euclidean
algorithm takes at most 2N steps to find gcd(m, n).

2.14 The Euclidean algorithm has been known for over 2000 years and has always been
a favorite among number theorists. After these many years, there is now a potential
competitor, invented by J. Stein in 1961. Stein’s algorithms is as follows: Determine
gcd(A, B) with A, B Ú 1.
STEP 1 Set A1 = A, B1 = B, C1 = 1
STEP 2 For n > 1, (1) If An = Bn, stop. gcd(A, B) = AnCn

(2) If An and Bn are both even, set An + 1 = An/2, Bn + 1 = Bn/2,
Cn + 1 = 2Cn

(3) If An is even and Bn is odd, set An + 1 = An/2, Bn + 1 = Bn,
Cn + 1 = Cn

(4) If An is odd and Bn is even, set An + 1 = An, Bn + 1 = Bn/2,
Cn + 1 = Cn

(5) If An and Bn are both odd, set An + 1 = �An - Bn � , Bn + 1 =
min (Bn, An), Cn + 1 = Cn

Continue to step n + 1.
a. To get a feel for the two algorithms, compute gcd(6150, 704) using both the Euclid-

ean and Stein’s algorithm.
b. What is the apparent advantage of Stein’s algorithm over the Euclidean algorithm?

2.15 a. Show that if Stein’s algorithm does not stop before the nth step, then

Cn + 1 * gcd(An + 1, Bn + 1) = Cn * gcd(An, Bn)

b. Show that if the algorithm does not stop before step (n - 1), then

An + 2Bn + 2 …
AnBn

2

c. Show that if 1 … A, B … 2N, then Stein’s algorithm takes at most 4N steps to find
gcd(m, n). Thus, Stein’s algorithm works in roughly the same number of steps as
the Euclidean algorithm.

d. Demonstrate that Stein’s algorithm does indeed return gcd(A, B).
2.16 Using the extended Euclidean algorithm, find the multiplicative inverse of

a. 135 mod 61
b. 7465 mod 2464
c. 42828 mod 6407

2.17 The purpose of this problem is to determine how many prime numbers there
are. Suppose there are a total of n prime numbers, and we list these in order:
p1 = 2 6 p2 = 3 6 p3 = 5 6 c 6 pn.
a. Define X = 1 + p1p2 c pn. That is, X is equal to one plus the product of all the

primes. Can we find a prime number Pm that divides X?
b. What can you say about m?
c. Deduce that the total number of primes cannot be finite.
d. Show that Pn + 1 … 1 + p1p2 c pn.

2.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 81

2.18 The purpose of this problem is to demonstrate that the probability that two random
numbers are relatively prime is about 0.6.
a. Let P = Pr[gcd(a, b) = 1]. Show that P = Pr[gcd(a, b) = d] = P/d2. Hint:

Consider the quantity gcd aa
d

,
b
d
b .

b. The sum of the result of part (a) over all possible values of d is 1. That is
Σd Ú1Pr[gcd(a, b) = d] = 1. Use this equality to determine the value of P. Hint:

Use the identity a
∞

i=1

1

i2
=
p2

6
.

2.19 Why is gcd(n, n + 1) = 1 for two consecutive integers n and n + 1?
2.20 Using Fermat’s theorem, find 4 2 2 5 mod 13.
2.21 Use Fermat’s theorem to find a number a between 0 and 92 with a congruent to 71013

modulo 93.
2.22 Use Fermat’s theorem to find a number x between 0 and 37 with x 7 3 congruent to 4

modulo 37. (You should not need to use any brute-force searching.)
2.23 Use Euler’s theorem to find a number a between 0 and 9 such that a is congruent to

9 1 0 1 modulo 10. (Note: This is the same as the last digit of the decimal expansion of
9 1 0 0.)

2.24 Use Euler’s theorem to find a number x between 0 and 14 with x 6 1 congruent to 7
modulo 15. (You should not need to use any brute-force searching.)

2.25 Notice in Table 2.6 that f(n) is even for n 7 2. This is true for all n 7 2. Give a con-
cise argument why this is so.

2.26 Prove the following: If p is prime, then f(pi) = pi - pi- 1. Hint: What numbers have
a factor in common with pi?

2.27 It can be shown (see any book on number theory) that if gcd(m, n) = 1 then
f(mn) = f(m)f(n). Using this property, the property developed in the preceding
problem, and the property that f(p) = p - 1 for p prime, it is straightforward to
determine the value of f(n) for any n. Determine the following:
a. f(29) b. f(51) c. f(455) d. f(616)

2.28 It can also be shown that for arbitrary positive integer a, f(a) is given by

f(a) = q
t

i=1
[pi

ai - 1(pi - 1)]

where a is given by Equation (2.9), namely: a = P1a1P2a2 c Ptat. Demonstrate this result.
2.29 Consider the function: f(n) = number of elements in the set {a: 0 … a 6 n and

gcd(a, n) = 1}. What is this function?
2.30 Although ancient Chinese mathematicians did good work coming up with their

remainder theorem, they did not always get it right. They had a test for primality. The
test said that n is prime if and only if n divides (2n - 2).
a. Give an example that satisfies the condition using an odd prime.
b. The condition is obviously true for n = 2. Prove that the condition is true if n is an

odd prime (proving the if condition).
c. Give an example of an odd n that is not prime and that does not satisfy the condi-

tion. You can do this with nonprime numbers up to a very large value. This misled
the Chinese mathematicians into thinking that if the condition is true then n is prime.

d. Unfortunately, the ancient Chinese never tried n = 341, which is nonprime
(341 = 11 * 31), yet 341 divides 2341 - 2 without remainder. Demonstrate that
2341 K 2 (mod 341) (disproving the only if condition). Hint: It is not necessary to
calculate 2341; play around with the congruences instead.

82 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

2.31 Show that, if n is an odd composite integer, then the Miller–Rabin test will return
inconclusive for a = 1 and a = (n - 1).

2.32 If n is composite and passes the Miller–Rabin test for the base a, then n is called
a strong pseudoprime to the base a. Show that 2047 is a strong pseudoprime to the
base 2.

2.33 A common formulation of the Chinese remainder theorem (CRT) is as follows: Let
m1, c , mk be integers that are pairwise relatively prime for 1 … i, j … k, and i ≠ j.
Define M to be the product of all the mi>s. Let a1, c , ak be integers. Then the set of
congruences:

x K a1(mod m1)
x K a2(mod m2)

~

~

~

x K ak(mod mk)

has a unique solution modulo M. Show that the theorem stated in this form is true.
2.34 The example used by Sun-Tsu to illustrate the CRT was

x K 2 (mod 3); x K 3 (mod 5); x K 2 (mod 7)

Solve for x.
2.35 Six professors begin courses on Monday, Tuesday, Wednesday, Thursday, Friday,

and Saturday, respectively, and announce their intentions of lecturing at intervals of
3, 2, 5, 6, 1, and 4 days, respectively. The regulations of the university forbid Sunday
lectures (so that a Sunday lecture must be omitted). When first will all six professors
find themselves compelled to omit a lecture? Hint: Use the CRT.

2.36 Find all primitive roots of 37.
2.37 Given 5 as a primitive root of 23, construct a table of discrete logarithms, and use it to

solve the following congruences.
a. 3x5 K 2 (mod 23)
b. 7x10 + 1 K 0 (mod 23)
c. 5x K 6 (mod 23)

Programming Problems

2.1 Write a computer program that implements fast exponentiation (successive squaring)
modulo n.

2.2 Write a computer program that implements the Miller–Rabin algorithm for a user-
specified n. The program should allow the user two choices: (1) specify a possible
witness a to test using the Witness procedure or (2) specify a number s of random
witnesses for the Miller–Rabin test to check.

APPENDIX 2A THE MEANING OF MOD

The operator mod is used in this book and in the literature in two different ways: as
a binary operator and as a congruence relation. This appendix explains the distinc-
tion and precisely defines the notation used in this book regarding parentheses. This
notation is common but, unfortunately, not universal.

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APPENDIX 2A / THE MEANING OF MOD 83

The Binary Operator mod

If a is an integer and n is a positive integer, we define a mod n to be the remainder
when a is divided by n. The integer n is called the modulus, and the remainder is
called the residue. Thus, for any integer a, we can always write

a = :a/n; * n + (a mod n)
Formally, we define the operator mod as

a mod n = a - :a/n; * n for n ≠ 0
As a binary operation, mod takes two integer arguments and returns the re-

mainder. For example, 7 mod 3 = 1. The arguments may be integers, integer vari-
ables, or integer variable expressions. For example, all of the following are valid,
with the obvious meanings:

7 mod 3

7 mod m

x mod 3

x mod m

(x2 + y + 1) mod (2m + n)

where all of the variables are integers. In each case, the left-hand term is divided by
the right-hand term, and the resulting value is the remainder. Note that if either the
left- or right-hand argument is an expression, the expression is parenthesized. The
operator mod is not inside parentheses.

In fact, the mod operation also works if the two arguments are arbitrary real num-
bers, not just integers. In this book, we are concerned only with the integer operation.

The Congruence Relation mod

As a congruence relation, mod expresses that two arguments have the same remain-
der with respect to a given modulus. For example, 7 K 4 (mod 3) expresses the
fact that both 7 and 4 have a remainder of 1 when divided by 3. The following two
expressions are equivalent:

a K b (mod m) 3 a mod m = b mod m

Another way of expressing it is to say that the expression a K b (mod m) is the
same as saying that a - b is an integral multiple of m. Again, all the arguments may
be integers, integer variables, or integer variable expressions. For example, all of
the following are valid, with the obvious meanings:

7 K 4 (mod 3)
x K y (mod m)
(x2 + y + 1) K (a + 1)(mod [m + n])

where all of the variables are integers. Two conventions are used. The congruence
sign is K . The modulus for the relation is defined by placing the mod operator fol-
lowed by the modulus in parentheses.

84 CHAPTER 2 / INTRODUCTION TO NUMBER THEORY

The congruence relation is used to define residue classes. Those numbers that
have the same remainder r when divided by m form a residue class (mod m). There
are m residue classes (mod m). For a given remainder r, the residue class to which it
belongs consists of the numbers

r, r { m, r { 2m, c

According to our definition, the congruence

a K b (mod m)

signifies that the numbers a and b differ by a multiple of m. Consequently, the con-
gruence can also be expressed in the terms that a and b belong to the same residue
class (mod m).

85

PART TWO: SYMMETRIC CIPHERS

CHAPTER

Classical Encryption Techniques
3.1 Symmetric Cipher Model

Cryptography
Cryptanalysis and Brute-Force Attack

3.2 Substitution Techniques

Caesar Cipher
Monoalphabetic Ciphers
Playfair Cipher
Hill Cipher
Polyalphabetic Ciphers
One-Time Pad

3.3 Transposition Techniques

3.4 Rotor Machines

3.5 Steganography

3.6 Key Terms, Review Questions, and Problems

86 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

Symmetric encryption, also referred to as conventional encryption or single-key
encryption, was the only type of encryption in use prior to the development of public-
key encryption in the 1970s. It remains by far the most widely used of the two types
of encryption. Part One examines a number of symmetric ciphers. In this chapter, we
begin with a look at a general model for the symmetric encryption process; this will
enable us to understand the context within which the algorithms are used. Next, we
examine a variety of algorithms in use before the computer era. Finally, we look briefly
at a different approach known as steganography. Chapters 4 and 6 introduce the two
most widely used symmetric cipher: DES and AES.

Before beginning, we define some terms. An original message is known as the
plaintext, while the coded message is called the ciphertext. The process of convert-
ing from plaintext to ciphertext is known as enciphering or encryption; restoring the
plaintext from the ciphertext is deciphering or decryption. The many schemes used
for encryption constitute the area of study known as cryptography. Such a scheme
is known as a cryptographic system or a cipher. Techniques used for deciphering a
message without any knowledge of the enciphering details fall into the area of crypt-
analysis. Cryptanalysis is what the layperson calls “breaking the code.” The areas of
cryptography and cryptanalysis together are called cryptology.

3.1 SYMMETRIC CIPHER MODEL

A symmetric encryption scheme has five ingredients (Figure 3.1):

■ Plaintext: This is the original intelligible message or data that is fed into the
algorithm as input.

■ Encryption algorithm: The encryption algorithm performs various substitu-
tions and transformations on the plaintext.

■ Secret key: The secret key is also input to the encryption algorithm. The key is
a value independent of the plaintext and of the algorithm. The algorithm will
produce a different output depending on the specific key being used at the
time. The exact substitutions and transformations performed by the algorithm
depend on the key.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Present an overview of the main concepts of symmetric cryptography.

◆ Explain the difference between cryptanalysis and brute-force attack.

◆ Understand the operation of a monoalphabetic substitution cipher.

◆ Understand the operation of a polyalphabetic cipher.

◆ Present an overview of the Hill cipher.

◆ Describe the operation of a rotor machine.

3.1 / SYMMETRIC CIPHER MODEL 87

■ Ciphertext: This is the scrambled message produced as output. It depends on
the plaintext and the secret key. For a given message, two different keys will
produce two different ciphertexts. The ciphertext is an apparently random
stream of data and, as it stands, is unintelligible.

■ Decryption algorithm: This is essentially the encryption algorithm run in
reverse. It takes the ciphertext and the secret key and produces the original
plaintext.

There are two requirements for secure use of conventional encryption:

1. We need a strong encryption algorithm. At a minimum, we would like the algo-
rithm to be such that an opponent who knows the algorithm and has access to
one or more ciphertexts would be unable to decipher the ciphertext or figure
out the key. This requirement is usually stated in a stronger form: The oppo-
nent should be unable to decrypt ciphertext or discover the key even if he or
she is in possession of a number of ciphertexts together with the plaintext that
produced each ciphertext.

2. Sender and receiver must have obtained copies of the secret key in a secure
fashion and must keep the key secure. If someone can discover the key and
knows the algorithm, all communication using this key is readable.

We assume that it is impractical to decrypt a message on the basis of the
ciphertext plus knowledge of the encryption/decryption algorithm. In other words,
we do not need to keep the algorithm secret; we need to keep only the key secret.
This feature of symmetric encryption is what makes it feasible for widespread use.
The fact that the algorithm need not be kept secret means that manufacturers can
and have developed low-cost chip implementations of data encryption algorithms.
These chips are widely available and incorporated into a number of products. With
the use of symmetric encryption, the principal security problem is maintaining the
secrecy of the key.

Let us take a closer look at the essential elements of a symmetric encryp-
tion scheme, using Figure 3.2. A source produces a message in plaintext,
X = [X1, X2, c , XM]. The M elements of X are letters in some finite alphabet.
Traditionally, the alphabet usually consisted of the 26 capital letters. Nowadays,

Figure 3.1 Simplified Model of Symmetric Encryption

Plaintext
input

Y = E(K, X ) X = D(K, Y )

X

KK

Transmitted
ciphertext

Plaintext
output

Secret key shared by
sender and recipient

Secret key shared by
sender and recipient

Encryption algorithm
(e.g., AES)

Decryption algorithm
(reverse of encryption

algorithm)

88 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

the binary alphabet {0, 1} is typically used. For encryption, a key of the form
K = [K1, K2, c , KJ] is generated. If the key is generated at the message source,
then it must also be provided to the destination by means of some secure channel.
Alternatively, a third party could generate the key and securely deliver it to both
source and destination.

With the message X and the encryption key K as input, the encryption algo-
rithm forms the ciphertext Y = [Y1, Y2, c , YN]. We can write this as

Y = E(K, X)

This notation indicates that Y is produced by using encryption algorithm E as a
function of the plaintext X, with the specific function determined by the value of
the key K.

The intended receiver, in possession of the key, is able to invert the
transformation:

X = D(K, Y)

An opponent, observing Y but not having access to K or X, may attempt to
recover X or K or both X and K. It is assumed that the opponent knows the encryp-
tion (E) and decryption (D) algorithms. If the opponent is interested in only this
particular message, then the focus of the effort is to recover X by generating a plain-
text estimate Xn . Often, however, the opponent is interested in being able to read
future messages as well, in which case an attempt is made to recover K by generat-
ing an estimate Kn .

Figure 3.2 Model of Symmetric Cryptosystem

Message
source

Cryptanalyst

Key
source

Destination
X X

X

K

Y = E(K, X )

Secure channel

K

Encryption
algorithm

Decryption
algorithm

3.1 / SYMMETRIC CIPHER MODEL 89

Cryptography

Cryptographic systems are characterized along three independent dimensions:

1. The type of operations used for transforming plaintext to ciphertext. All
encryption algorithms are based on two general principles: substitution,
in which each element in the plaintext (bit, letter, group of bits or letters)
is mapped into another element, and transposition, in which elements
in the plaintext are rearranged. The fundamental requirement is that no
information be lost (i.e., that all operations are reversible). Most systems,
referred to as product systems, involve multiple stages of substitutions and
transpositions.

2. The number of keys used. If both sender and receiver use the same key, the
system is referred to as symmetric, single-key, secret-key, or conventional
encryption. If the sender and receiver use different keys, the system is referred
to as asymmetric, two-key, or public-key encryption.

3. The way in which the plaintext is processed. A block cipher processes the input
one block of elements at a time, producing an output block for each input
block. A stream cipher processes the input elements continuously, producing
output one element at a time, as it goes along.

Cryptanalysis and Brute-Force Attack

Typically, the objective of attacking an encryption system is to recover the key in
use rather than simply to recover the plaintext of a single ciphertext. There are two
general approaches to attacking a conventional encryption scheme:

■ Cryptanalysis: Cryptanalytic attacks rely on the nature of the algorithm plus
perhaps some knowledge of the general characteristics of the plaintext or even
some sample plaintext–ciphertext pairs. This type of attack exploits the charac-
teristics of the algorithm to attempt to deduce a specific plaintext or to deduce
the key being used.

■ Brute-force attack: The attacker tries every possible key on a piece of cipher-
text until an intelligible translation into plaintext is obtained. On average, half
of all possible keys must be tried to achieve success.

If either type of attack succeeds in deducing the key, the effect is catastrophic:
All future and past messages encrypted with that key are compromised.

We first consider cryptanalysis and then discuss brute-force attacks.
Table 3.1 summarizes the various types of cryptanalytic attacks based on the

amount of information known to the cryptanalyst. The most difficult problem is
presented when all that is available is the ciphertext only. In some cases, not even
the encryption algorithm is known, but in general, we can assume that the opponent
does know the algorithm used for encryption. One possible attack under these cir-
cumstances is the brute-force approach of trying all possible keys. If the key space
is very large, this becomes impractical. Thus, the opponent must rely on an analysis
of the ciphertext itself, generally applying various statistical tests to it. To use this

90 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

approach, the opponent must have some general idea of the type of plaintext that
is concealed, such as English or French text, an EXE file, a Java source listing, an
accounting file, and so on.

The ciphertext-only attack is the easiest to defend against because the oppo-
nent has the least amount of information to work with. In many cases, however,
the analyst has more information. The analyst may be able to capture one or more
plaintext messages as well as their encryptions. Or the analyst may know that certain
plaintext patterns will appear in a message. For example, a file that is encoded in the
Postscript format always begins with the same pattern, or there may be a standard-
ized header or banner to an electronic funds transfer message, and so on. All these
are examples of known plaintext. With this knowledge, the analyst may be able to
deduce the key on the basis of the way in which the known plaintext is transformed.

Closely related to the known-plaintext attack is what might be referred to as a
probable-word attack. If the opponent is working with the encryption of some gen-
eral prose message, he or she may have little knowledge of what is in the message.
However, if the opponent is after some very specific information, then parts of the
message may be known. For example, if an entire accounting file is being transmit-
ted, the opponent may know the placement of certain key words in the header of the
file. As another example, the source code for a program developed by Corporation
X might include a copyright statement in some standardized position.

If the analyst is able somehow to get the source system to insert into the sys-
tem a message chosen by the analyst, then a chosen-plaintext attack is possible.
An example of this strategy is differential cryptanalysis, explored in Appendix S.

Type of Attack Known to Cryptanalyst

Ciphertext Only ■ Encryption algorithm
■ Ciphertext

Known Plaintext ■ Encryption algorithm
■ Ciphertext
■ One or more plaintext–ciphertext pairs formed with the secret key

Chosen Plaintext ■ Encryption algorithm
■ Ciphertext
■ Plaintext message chosen by cryptanalyst, together with its corresponding

ciphertext generated with the secret key

Chosen Ciphertext ■ Encryption algorithm
■ Ciphertext
■ Ciphertext chosen by cryptanalyst, together with its corresponding decrypted

plaintext generated with the secret key

Chosen Text ■ Encryption algorithm
■ Ciphertext
■ Plaintext message chosen by cryptanalyst, together with its corresponding

ciphertext generated with the secret key
■ Ciphertext chosen by cryptanalyst, together with its corresponding decrypted

plaintext generated with the secret key

Table 3.1 Types of Attacks on Encrypted Messages

3.1 / SYMMETRIC CIPHER MODEL 91

In general, if the analyst is able to choose the messages to encrypt, the analyst may
deliberately pick patterns that can be expected to reveal the structure of the key.

Table 3.1 lists two other types of attack: chosen ciphertext and chosen text.
These are less commonly employed as cryptanalytic techniques but are nevertheless
possible avenues of attack.

Only relatively weak algorithms fail to withstand a ciphertext-only attack.
Generally, an encryption algorithm is designed to withstand a known-plaintext
attack.

Two more definitions are worthy of note. An encryption scheme is
unconditionally secure if the ciphertext generated by the scheme does not contain
enough information to determine uniquely the corresponding plaintext, no matter
how much ciphertext is available. That is, no matter how much time an opponent
has, it is impossible for him or her to decrypt the ciphertext simply because the
required information is not there. With the exception of a scheme known as the
one-time pad (described later in this chapter), there is no encryption algorithm that
is unconditionally secure. Therefore, all that the users of an encryption algorithm
can strive for is an algorithm that meets one or both of the following criteria:

■ The cost of breaking the cipher exceeds the value of the encrypted information.

■ The time required to break the cipher exceeds the useful lifetime of the
information.

An encryption scheme is said to be computationally secure if either of the
foregoing two criteria are met. Unfortunately, it is very difficult to estimate the
amount of effort required to cryptanalyze ciphertext successfully.

All forms of cryptanalysis for symmetric encryption schemes are designed
to exploit the fact that traces of structure or pattern in the plaintext may survive
encryption and be discernible in the ciphertext. This will become clear as we exam-
ine various symmetric encryption schemes in this chapter. We will see in Part Two
that cryptanalysis for public-key schemes proceeds from a fundamentally different
premise, namely, that the mathematical properties of the pair of keys may make it
possible for one of the two keys to be deduced from the other.

A brute-force attack involves trying every possible key until an intelligible
translation of the ciphertext into plaintext is obtained. On average, half of all pos-
sible keys must be tried to achieve success. That is, if there are X different keys, on
average an attacker would discover the actual key after X/2 tries. It is important to
note that there is more to a brute-force attack than simply running through all pos-
sible keys. Unless known plaintext is provided, the analyst must be able to recognize
plaintext as plaintext. If the message is just plain text in English, then the result pops
out easily, although the task of recognizing English would have to be automated. If
the text message has been compressed before encryption, then recognition is more
difficult. And if the message is some more general type of data, such as a numeri-
cal file, and this has been compressed, the problem becomes even more difficult to
automate. Thus, to supplement the brute-force approach, some degree of knowl-
edge about the expected plaintext is needed, and some means of automatically dis-
tinguishing plaintext from garble is also needed.

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92 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

3.2 SUBSTITUTION TECHNIQUES

In this section and the next, we examine a sampling of what might be called classical
encryption techniques. A study of these techniques enables us to illustrate the basic
approaches to symmetric encryption used today and the types of cryptanalytic at-
tacks that must be anticipated.

The two basic building blocks of all encryption techniques are substitution
and transposition. We examine these in the next two sections. Finally, we discuss a
system that combines both substitution and transposition.

A substitution technique is one in which the letters of plaintext are replaced
by other letters or by numbers or symbols.1 If the plaintext is viewed as a sequence
of bits, then substitution involves replacing plaintext bit patterns with ciphertext bit
patterns.

Caesar Cipher

The earliest known, and the simplest, use of a substitution cipher was by Julius
Caesar. The Caesar cipher involves replacing each letter of the alphabet with the
letter standing three places further down the alphabet. For example,

plain: meet me after the toga party
cipher: PHHW PH DIWHU WKH WRJD SDUWB

Note that the alphabet is wrapped around, so that the letter following Z is A.
We can define the transformation by listing all possibilities, as follows:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y z
cipher: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

Let us assign a numerical equivalent to each letter:

a b c d e f g h i j k l m

0 1 2 3 4 5 6 7 8 9 10 11 12

n o p q r s t u v w x y z

13 14 15 16 17 18 19 20 21 22 23 24 25

Then the algorithm can be expressed as follows. For each plaintext letter p, substi-
tute the ciphertext letter C:2

C = E(3, p) = (p + 3) mod 26

A shift may be of any amount, so that the general Caesar algorithm is

C = E(k, p) = (p + k) mod 26 (3.1)

1When letters are involved, the following conventions are used in this book. Plaintext is always in
lowercase; ciphertext is in uppercase; key values are in italicized lowercase.
2We define a mod n to be the remainder when a is divided by n. For example, 11 mod 7 = 4. See Chapter 2
for a further discussion of modular arithmetic.

3.2 / SUBSTITUTION TECHNIQUES 93

where k takes on a value in the range 1 to 25. The decryption algorithm is simply

p = D(k, C) = (C - k) mod 26 (3.2)

If it is known that a given ciphertext is a Caesar cipher, then a brute-force
cryptanalysis is easily performed: simply try all the 25 possible keys. Figure 3.3
shows the results of applying this strategy to the example ciphertext. In this case, the
plaintext leaps out as occupying the third line.

Three important characteristics of this problem enabled us to use a brute-
force cryptanalysis:

1. The encryption and decryption algorithms are known.

2. There are only 25 keys to try.

3. The language of the plaintext is known and easily recognizable.

In most networking situations, we can assume that the algorithms are known.
What generally makes brute-force cryptanalysis impractical is the use of an algo-
rithm that employs a large number of keys. For example, the triple DES algorithm,

Figure 3.3 Brute-Force Cryptanalysis of Caesar Cipher

PHHW PH DIWHU WKH WRJD SDUWB
KEY

1 oggv og chvgt vjg vqic rctva

2 nffu nf bgufs uif uphb qbsuz

3 meet me after the toga party

4 ldds ld zesdq sgd snfz ozqsx

5 kccr kc ydrcp rfc rmey nyprw

6 jbbq jb xcqbo qeb qldx mxoqv

7 iaap ia wbpan pda pkcw lwnpu

8 hzzo hz vaozm ocz ojbv kvmot

9 gyyn gy uznyl nby niau julns

10 fxxm fx tymxk max mhzt itkmr

11 ewwl ew sxlwj lzw lgys hsjlq

12 dvvk dv rwkvi kyv kfxr grikp

13 cuuj cu qvjuh jxu jewq fqhjo

14 btti bt puitg iwt idvp epgin

15 assh as othsf hvs hcuo dofhm

16 zrrg zr nsgre gur gbtn cnegl

17 yqqf yq mrfqd ftq fasm bmdfk

18 xppe xp lqepc esp ezrl alcej

19 wood wo kpdob dro dyqk zkbdi

20 vnnc vn jocna cqn cxpj yjach

21 ummb um inbmz bpm bwoi xizbg

22 tlla tl hmaly aol avnh whyaf

23 skkz sk glzkx znk zumg vgxze

24 rjjy rj fkyjw ymj ytlf ufwyd

25 qiix qi ejxiv xli xske tevxc

94 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

examined in Chapter 7, makes use of a 168-bit key, giving a key space of 2168 or
greater than 3.7 * 1050 possible keys.

The third characteristic is also significant. If the language of the plaintext is
unknown, then plaintext output may not be recognizable. Furthermore, the input
may be abbreviated or compressed in some fashion, again making recognition dif-
ficult. For example, Figure 3.4 shows a portion of a text file compressed using an
algorithm called ZIP. If this file is then encrypted with a simple substitution cipher
(expanded to include more than just 26 alphabetic characters), then the plaintext
may not be recognized when it is uncovered in the brute-force cryptanalysis.

Monoalphabetic Ciphers

With only 25 possible keys, the Caesar cipher is far from secure. A dramatic increase
in the key space can be achieved by allowing an arbitrary substitution. Before pro-
ceeding, we define the term permutation. A permutation of a finite set of elements S
is an ordered sequence of all the elements of S, with each element appearing exactly
once. For example, if S = {a, b, c}, there are six permutations of S:

abc, acb, bac, bca, cab, cba

In general, there are n! permutations of a set of n elements, because the first
element can be chosen in one of n ways, the second in n - 1 ways, the third in n - 2
ways, and so on.

Recall the assignment for the Caesar cipher:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y z
cipher: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

If, instead, the “cipher” line can be any permutation of the 26 alphabetic characters,
then there are 26! or greater than 4 * 1026 possible keys. This is 10 orders of mag-
nitude greater than the key space for DES and would seem to eliminate brute-force
techniques for cryptanalysis. Such an approach is referred to as a monoalphabetic
substitution cipher, because a single cipher alphabet (mapping from plain alphabet
to cipher alphabet) is used per message.

There is, however, another line of attack. If the cryptanalyst knows the nature
of the plaintext (e.g., noncompressed English text), then the analyst can exploit the
regularities of the language. To see how such a cryptanalysis might proceed, we give
a partial example here that is adapted from one in [SINK09]. The ciphertext to be
solved is

Figure 3.4 Sample of Compressed Text

3.2 / SUBSTITUTION TECHNIQUES 95

UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ
VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX
EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ

As a first step, the relative frequency of the letters can be determined and
compared to a standard frequency distribution for English, such as is shown in
Figure 3.5 (based on [LEWA00]). If the message were long enough, this technique
alone might be sufficient, but because this is a relatively short message, we cannot
expect an exact match. In any case, the relative frequencies of the letters in the
ciphertext (in percentages) are as follows:

P 13.33 H 5.83 F 3.33 B 1.67 C 0.00

Z 11.67 D 5.00 W 3.33 G 1.67 K 0.00

S 8.33 E 5.00 Q 2.50 Y 1.67 L 0.00

U 8.33 V 4.17 T 2.50 I 0.83 N 0.00

O 7.50 X 4.17 A 1.67 J 0.83 R 0.00

M 6.67

Comparing this breakdown with Figure 3.5, it seems likely that cipher letters
P and Z are the equivalents of plain letters e and t, but it is not certain which is which.
The letters S, U, O, M, and H are all of relatively high frequency and probably

Figure 3.5 Relative Frequency of Letters in English Text

0

2

4

6

8

10

12

14

A

8.
16

7

1.
49

2

2.
78

2

4.
25

3

12
.7

02

2.
22

8

2.
01

5

6.
09

4 6
.9

96

0.
15

3 0.
77

2

4.
02

5

2.
40

6

6.
74

9 7.
50

7

1.
92

9

0.
09

5

5.
98

7

6.
32

7

9.
05

6

2.
75

8

0.
97

8

2.
36

0

0.
15

0

1.
97

4

0.
07

4

B C D E F G H I J K L M N

R
el

at
iv

e
fr

eq
ue

nc
y

(%
)

O P Q R S T U V W X Y Z

96 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

correspond to plain letters from the set {a, h, i, n, o, r, s}. The letters with the lowest
frequencies (namely, A, B, G, Y, I, J) are likely included in the set {b, j, k, q, v, x, z}.

There are a number of ways to proceed at this point. We could make some
tentative assignments and start to fill in the plaintext to see if it looks like a rea-
sonable “skeleton” of a message. A more systematic approach is to look for other
regularities. For example, certain words may be known to be in the text. Or we
could look for repeating sequences of cipher letters and try to deduce their plaintext
equivalents.

A powerful tool is to look at the frequency of two-letter combinations, known
as digrams. A table similar to Figure 3.5 could be drawn up showing the relative fre-
quency of digrams. The most common such digram is th. In our ciphertext, the most
common digram is ZW, which appears three times. So we make the correspondence
of Z with t and W with h. Then, by our earlier hypothesis, we can equate P with e.
Now notice that the sequence ZWP appears in the ciphertext, and we can translate
that sequence as “the.” This is the most frequent trigram (three-letter combination)
in English, which seems to indicate that we are on the right track.

Next, notice the sequence ZWSZ in the first line. We do not know that these
four letters form a complete word, but if they do, it is of the form th_t. If so, S
equates with a.

So far, then, we have

UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ
t a e e te a that e e a a
VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX

e t ta t ha e ee a e th t a
EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ
e e e tat e the t

Only four letters have been identified, but already we have quite a bit of the
message. Continued analysis of frequencies plus trial and error should easily yield a
solution from this point. The complete plaintext, with spaces added between words,
follows:

it was disclosed yesterday that several informal but
direct contacts have been made with political
representatives of the viet cong in moscow

Monoalphabetic ciphers are easy to break because they reflect the frequency
data of the original alphabet. A countermeasure is to provide multiple substi-
tutes, known as homophones, for a single letter. For example, the letter e could
be assigned a number of different cipher symbols, such as 16, 74, 35, and 21, with
each homophone assigned to a letter in rotation or randomly. If the number of
symbols assigned to each letter is proportional to the relative frequency of that let-
ter, then single-letter frequency information is completely obliterated. The great
mathematician Carl Friedrich Gauss believed that he had devised an unbreak-
able cipher using homophones. However, even with homophones, each element
of plaintext affects only one element of ciphertext, and multiple-letter patterns

3.2 / SUBSTITUTION TECHNIQUES 97

(e.g., digram frequencies) still survive in the ciphertext, making cryptanalysis rela-
tively straightforward.

Two principal methods are used in substitution ciphers to lessen the extent to
which the structure of the plaintext survives in the ciphertext: One approach is to
encrypt multiple letters of plaintext, and the other is to use multiple cipher alpha-
bets. We briefly examine each.

Playfair Cipher

The best-known multiple-letter encryption cipher is the Playfair, which treats di-
grams in the plaintext as single units and translates these units into ciphertext
digrams.3

The Playfair algorithm is based on the use of a 5 * 5 matrix of letters con-
structed using a keyword. Here is an example, solved by Lord Peter Wimsey in
Dorothy Sayers’s Have His Carcase:4

M O N A R

C H Y B D

E F G I/J K

L P Q S T

U V W X Z

In this case, the keyword is monarchy. The matrix is constructed by filling
in the letters of the keyword (minus duplicates) from left to right and from top to
bottom, and then filling in the remainder of the matrix with the remaining letters in
alphabetic order. The letters I and J count as one letter. Plaintext is encrypted two
letters at a time, according to the following rules:

1. Repeating plaintext letters that are in the same pair are separated with a filler
letter, such as x, so that balloon would be treated as ba lx lo on.

2. Two plaintext letters that fall in the same row of the matrix are each replaced
by the letter to the right, with the first element of the row circularly following
the last. For example, ar is encrypted as RM.

3. Two plaintext letters that fall in the same column are each replaced by the let-
ter beneath, with the top element of the column circularly following the last.
For example, mu is encrypted as CM.

4. Otherwise, each plaintext letter in a pair is replaced by the letter that lies in
its own row and the column occupied by the other plaintext letter. Thus, hs
becomes BP and ea becomes IM (or JM, as the encipherer wishes).

The Playfair cipher is a great advance over simple monoalphabetic ciphers.
For one thing, whereas there are only 26 letters, there are 26 * 26 = 676 digrams,

3This cipher was actually invented by British scientist Sir Charles Wheatstone in 1854, but it bears the
name of his friend Baron Playfair of St. Andrews, who championed the cipher at the British foreign office.
4The book provides an absorbing account of a probable-word attack.

98 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

so that identification of individual digrams is more difficult. Furthermore, the rela-
tive frequencies of individual letters exhibit a much greater range than that of
digrams, making frequency analysis much more difficult. For these reasons, the
Playfair cipher was for a long time considered unbreakable. It was used as the stan-
dard field system by the British Army in World War I and still enjoyed considerable
use by the U.S. Army and other Allied forces during World War II.

Despite this level of confidence in its security, the Playfair cipher is relatively
easy to break, because it still leaves much of the structure of the plaintext language
intact. A few hundred letters of ciphertext are generally sufficient.

One way of revealing the effectiveness of the Playfair and other ciphers is
shown in Figure 3.6. The line labeled plaintext plots a typical frequency distribution
of the 26 alphabetic characters (no distinction between upper and lower case) in
ordinary text. This is also the frequency distribution of any monoalphabetic substi-
tution cipher, because the frequency values for individual letters are the same, just
with different letters substituted for the original letters. The plot is developed in the
following way: The number of occurrences of each letter in the text is counted and
divided by the number of occurrences of the most frequently used letter. Using the
results of Figure 3.5, we see that e is the most frequently used letter. As a result, e
has a relative frequency of 1, t of 9.056/12.702 ≈ 0.72, and so on. The points on the
horizontal axis correspond to the letters in order of decreasing frequency.

Figure 3.6 also shows the frequency distribution that results when the text is
encrypted using the Playfair cipher. To normalize the plot, the number of occur-
rences of each letter in the ciphertext was again divided by the number of occur-
rences of e in the plaintext. The resulting plot therefore shows the extent to which
the frequency distribution of letters, which makes it trivial to solve substitution

Figure 3.6 Relative Frequency of Occurrence of Letters

0
1 2 3 4 5 6 1 7 8 9 10 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Plaintext

Playfair

Vigenère

Random polyalphabetic

Frequency ranked letters (decreasing frequency)

N
or

m
al

iz
ed

re
la

tiv
e

fr
eq

ue
nc

y

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

3.2 / SUBSTITUTION TECHNIQUES 99

ciphers, is masked by encryption. If the frequency distribution information were
totally concealed in the encryption process, the ciphertext plot of frequencies would
be flat, and cryptanalysis using ciphertext only would be effectively impossible. As
the figure shows, the Playfair cipher has a flatter distribution than does plaintext,
but nevertheless, it reveals plenty of structure for a cryptanalyst to work with. The
plot also shows the Vigenère cipher, discussed subsequently. The Hill and Vigenère
curves on the plot are based on results reported in [SIMM93].

Hill Cipher5

Another interesting multiletter cipher is the Hill cipher, developed by the math-
ematician Lester Hill in 1929.

CONCEPTS FROM LINEAR ALGEBRA Before describing the Hill cipher, let us briefly
review some terminology from linear algebra. In this discussion, we are concerned
with matrix arithmetic modulo 26. For the reader who needs a refresher on matrix
multiplication and inversion, see Appendix E.

We define the inverse M-1 of a square matrix M by the equation M(M-1) =
M-1M = I, where I is the identity matrix. I is a square matrix that is all zeros except
for ones along the main diagonal from upper left to lower right. The inverse of a
matrix does not always exist, but when it does, it satisfies the preceding equation.
For example,

A = ¢ 5 8
17 3

≤ A-1 mod 26 = ¢9 2
1 15

≤
AA-1 = ¢ (5 * 9) + (8 * 1) (5 * 2) + (8 * 15)

(17 * 9) + (3 * 1) (17 * 2) + (3 * 15)
≤

= ¢ 53 130
156 79

≤ mod 26 = ¢1 0
0 1

≤
To explain how the inverse of a matrix is computed, we begin with the concept

of determinant. For any square matrix (m * m), the determinant equals the sum of
all the products that can be formed by taking exactly one element from each row
and exactly one element from each column, with certain of the product terms pre-
ceded by a minus sign. For a 2 * 2 matrix,

¢k11 k12
k21 k22

≤
the determinant is k11k22 - k12k21. For a 3 * 3 matrix, the value of the determinant
is k11k22k33 + k21k32k13 + k31k12k23 - k31k22k13 - k21k12k33 - k11k32k23. If a square

5This cipher is somewhat more difficult to understand than the others in this chapter, but it illustrates an
important point about cryptanalysis that will be useful later on. This subsection can be skipped on a first
reading.

100 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

matrix A has a nonzero determinant, then the inverse of the matrix is computed
as [A-1]ij = (det A)-1(-1)i+ j(Dji), where (Dji) is the subdeterminant formed by
deleting the jth row and the ith column of A, det(A) is the determinant of A, and
(det A)-1 is the multiplicative inverse of (det A) mod 26.

Continuing our example,

det ¢ 5 8
17 3

≤ = (5 * 3) - (8 * 17) = -121 mod 26 = 9
We can show that 9-1 mod 26 = 3, because 9 * 3 = 27 mod 26 = 1 (see

Chapter 2 or Appendix E). Therefore, we compute the inverse of A as

A = ¢ 5 8
17 3

≤
A-1 mod 26 = 3¢ 3 -8

-17 5
≤ = 3¢3 18

9 5
≤ = ¢ 9 54

27 15
≤ = ¢9 2

1 15
≤

THE HILL ALGORITHM This encryption algorithm takes m successive plaintext let-
ters and substitutes for them m ciphertext letters. The substitution is determined
by m linear equations in which each character is assigned a numerical value
(a = 0, b = 1, c , z = 25). For m = 3, the system can be described as

c1 = (k11p1 + k21p2 + k31p3) mod 26

c2 = (k12p1 + k22p2 + k32p3) mod 26

c3 = (k13p1 + k23p2 + k33p3) mod 26

This can be expressed in terms of row vectors and matrices:6

(c1 c2 c3) = (p1 p2 p3)£k11 k12 k13k21 k22 k23
k31 k32 k33

≥ mod 26
or

C = PK mod 26

where C and P are row vectors of length 3 representing the plaintext and ciphertext,
and K is a 3 * 3 matrix representing the encryption key. Operations are performed
mod 26.

6Some cryptography books express the plaintext and ciphertext as column vectors, so that the column
vector is placed after the matrix rather than the row vector placed before the matrix. Sage uses row vec-
tors, so we adopt that convention.

Hiva-Network.Com

3.2 / SUBSTITUTION TECHNIQUES 101

For example, consider the plaintext “paymoremoney” and use the encryption key

K = £17 17 521 18 21
2 2 19

≥
The first three letters of the plaintext are represented by the vector (15 0 24).
Then (15 0 24)K = (303 303 531) mod 26 = (17 17 11) = RRL. Continuing in this
fashion, the ciphertext for the entire plaintext is RRLMWBKASPDH.

Decryption requires using the inverse of the matrix K. We can compute det
K = 23, and therefore, (det K)-1 mod 26 = 17. We can then compute the inverse as7

K-1 = £ 4 9 1515 17 6
24 0 17

≥
This is demonstrated as

£17 17 521 18 21
2 2 19

≥£ 4 9 1515 17 6
24 0 17

≥ = £443 442 442858 495 780
494 52 365

≥ mod 26 = £1 0 00 1 0
0 0 1

≥
It is easily seen that if the matrix K-1 is applied to the ciphertext, then the

plaintext is recovered.
In general terms, the Hill system can be expressed as

C = E(K, P) = PK mod 26

P = D(K, C) = CK-1 mod 26 = PKK-1 = P

As with Playfair, the strength of the Hill cipher is that it completely hides
single-letter frequencies. Indeed, with Hill, the use of a larger matrix hides more
frequency information. Thus, a 3 * 3 Hill cipher hides not only single-letter but
also two-letter frequency information.

Although the Hill cipher is strong against a ciphertext-only attack, it is easily
broken with a known plaintext attack. For an m * m Hill cipher, suppose we have m
plaintext–ciphertext pairs, each of length m. We label the pairs Pj = (p1jp1j c pmj)
and Cj = (c1jc1j c cmj) such that Cj = PjK for 1 … j … m and for some unknown
key matrix K. Now define two m * m matrices X = (pij) and Y = (cij). Then we
can form the matrix equation Y = XK. If X has an inverse, then we can determine
K = X-1Y. If X is not invertible, then a new version of X can be formed with addi-
tional plaintext–ciphertext pairs until an invertible X is obtained.

Consider this example. Suppose that the plaintext “hillcipher” is encrypted
using a 2 * 2 Hill cipher to yield the ciphertext HCRZSSXNSP. Thus, we know
that (7 8)K mod 26 = (7 2); (11 11)K mod 26 = (17 25); and so on. Using
the first two plaintext-ciphertext pairs, we have

7The calculations for this example are provided in detail in Appendix E.

102 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

¢ 7 2
17 25

≤ = ¢ 7 8
11 11

≤K mod 26
The inverse of X can be computed:

¢ 7 8
11 11

≤-1 = ¢25 22
1 23

≤
so

K = ¢25 22
1 23

≤ ¢ 7 2
17 25

≤ = ¢549 600
398 577

≤ mod 26 = ¢3 2
8 5

≤
This result is verified by testing the remaining plaintext–ciphertext pairs.

Polyalphabetic Ciphers

Another way to improve on the simple monoalphabetic technique is to use differ-
ent monoalphabetic substitutions as one proceeds through the plaintext message.
The general name for this approach is polyalphabetic substitution cipher. All these
techniques have the following features in common:

1. A set of related monoalphabetic substitution rules is used.

2. A key determines which particular rule is chosen for a given transformation.

VIGENÈRE CIPHER The best known, and one of the simplest, polyalphabetic ciphers
is the Vigenère cipher. In this scheme, the set of related monoalphabetic substitu-
tion rules consists of the 26 Caesar ciphers with shifts of 0 through 25. Each cipher is
denoted by a key letter, which is the ciphertext letter that substitutes for the plain-
text letter a. Thus, a Caesar cipher with a shift of 3 is denoted by the key value 3.8

We can express the Vigenère cipher in the following manner. Assume a
sequence of plaintext letters P = p0, p1, p2, c , pn - 1 and a key consisting of the
sequence of letters K = k0, k1, k2, c , km - 1, where typically m 6 n. The sequence
of ciphertext letters C = C0, C1, C2, c , Cn - 1 is calculated as follows:

C = C0, C1, C2, c , Cn - 1 = E(K, P) = E[(k0, k1, k2, c , km - 1), (p0, p1, p2, c , pn - 1)]

= (p0 + k0) mod 26, (p1 + k1) mod 26, c ,(pm - 1 + km - 1) mod 26,

(pm + k0) mod 26, (pm + 1 + k1) mod 26, c , (p2m - 1 + km - 1) mod 26, c

Thus, the first letter of the key is added to the first letter of the plaintext, mod 26,
the second letters are added, and so on through the first m letters of the plaintext.
For the next m letters of the plaintext, the key letters are repeated. This process

8To aid in understanding this scheme and also to aid in it use, a matrix known as the Vigenère tableau is
often used. This tableau is discussed in a document at box.com/Crypto7e.

3.2 / SUBSTITUTION TECHNIQUES 103

continues until all of the plaintext sequence is encrypted. A general equation of the
encryption process is

Ci = (pi + ki mod m) mod 26 (3.3)

Compare this with Equation (3.1) for the Caesar cipher. In essence, each plain-
text character is encrypted with a different Caesar cipher, depending on the corre-
sponding key character. Similarly, decryption is a generalization of Equation (3.2):

pi = (Ci - ki mod m) mod 26 (3.4)

To encrypt a message, a key is needed that is as long as the message. Usually,
the key is a repeating keyword. For example, if the keyword is deceptive, the mes-
sage “we are discovered save yourself” is encrypted as

key: deceptivedeceptivedeceptive
plaintext: wearediscoveredsaveyourself
ciphertext: ZICVTWQNGRZGVTWAVZHCQYGLMGJ

Expressed numerically, we have the following result.

key 3 4 2 4 15 19 8 21 4 3 4 2 4 15

plaintext 22 4 0 17 4 3 8 18 2 14 21 4 17 4

ciphertext 25 8 2 21 19 22 16 13 6 17 25 6 21 19

key 19 8 21 4 3 4 2 4 15 19 8 21 4

plaintext 3 18 0 21 4 24 14 20 17 18 4 11 5

ciphertext 22 0 21 25 7 2 16 24 6 11 12 6 9

The strength of this cipher is that there are multiple ciphertext letters for
each plaintext letter, one for each unique letter of the keyword. Thus, the letter fre-
quency information is obscured. However, not all knowledge of the plaintext struc-
ture is lost. For example, Figure 3.6 shows the frequency distribution for a Vigenère
cipher with a keyword of length 9. An improvement is achieved over the Playfair
cipher, but considerable frequency information remains.

It is instructive to sketch a method of breaking this cipher, because the method
reveals some of the mathematical principles that apply in cryptanalysis.

First, suppose that the opponent believes that the ciphertext was encrypted
using either monoalphabetic substitution or a Vigenère cipher. A simple test can
be made to make a determination. If a monoalphabetic substitution is used, then
the statistical properties of the ciphertext should be the same as that of the lan-
guage of the plaintext. Thus, referring to Figure 3.5, there should be one cipher let-
ter with a relative frequency of occurrence of about 12.7%, one with about 9.06%,
and so on. If only a single message is available for analysis, we would not expect
an exact match of this small sample with the statistical profile of the plaintext lan-
guage. Nevertheless, if the correspondence is close, we can assume a monoalpha-
betic substitution.

104 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

If, on the other hand, a Vigenère cipher is suspected, then progress depends on
determining the length of the keyword, as will be seen in a moment. For now, let us
concentrate on how the keyword length can be determined. The important insight
that leads to a solution is the following: If two identical sequences of plaintext let-
ters occur at a distance that is an integer multiple of the keyword length, they will
generate identical ciphertext sequences. In the foregoing example, two instances
of the sequence “red” are separated by nine character positions. Consequently, in
both cases, r is encrypted using key letter e, e is encrypted using key letter p, and d
is encrypted using key letter t. Thus, in both cases, the ciphertext sequence is VTW.
We indicate this above by underlining the relevant ciphertext letters and shading
the relevant ciphertext numbers.

An analyst looking at only the ciphertext would detect the repeated sequences
VTW at a displacement of 9 and make the assumption that the keyword is either
three or nine letters in length. The appearance of VTW twice could be by chance
and may not reflect identical plaintext letters encrypted with identical key letters.
However, if the message is long enough, there will be a number of such repeated
ciphertext sequences. By looking for common factors in the displacements of the vari-
ous sequences, the analyst should be able to make a good guess of the keyword length.

Solution of the cipher now depends on an important insight. If the keyword
length is m, then the cipher, in effect, consists of m monoalphabetic substitution
ciphers. For example, with the keyword DECEPTIVE, the letters in positions 1, 10,
19, and so on are all encrypted with the same monoalphabetic cipher. Thus, we can
use the known frequency characteristics of the plaintext language to attack each of
the monoalphabetic ciphers separately.

The periodic nature of the keyword can be eliminated by using a nonrepeating
keyword that is as long as the message itself. Vigenère proposed what is referred to
as an autokey system, in which a keyword is concatenated with the plaintext itself to
provide a running key. For our example,

key: deceptivewearediscoveredsav
plaintext: wearediscoveredsaveyourself
ciphertext: ZICVTWQNGKZEIIGASXSTSLVVWLA

Even this scheme is vulnerable to cryptanalysis. Because the key and the
plaintext share the same frequency distribution of letters, a statistical technique can
be applied. For example, e enciphered by e, by Figure 3.5, can be expected to occur
with a frequency of (0.127)2 ≈ 0.016, whereas t enciphered by t would occur only
about half as often. These regularities can be exploited to achieve successful
cryptanalysis.9

VERNAM CIPHER The ultimate defense against such a cryptanalysis is to choose a
keyword that is as long as the plaintext and has no statistical relationship to it. Such
a system was introduced by an AT&T engineer named Gilbert Vernam in 1918.

9Although the techniques for breaking a Vigenère cipher are by no means complex, a 1917 issue of
Scientific American characterized this system as “impossible of translation.” This is a point worth remem-
bering when similar claims are made for modern algorithms.

3.2 / SUBSTITUTION TECHNIQUES 105

His system works on binary data (bits) rather than letters. The system can be
expressed succinctly as follows (Figure 3.7):

ci = pi⊕ ki

where

pi = ith binary digit of plaintext
ki = ith binary digit of key
ci = ith binary digit of ciphertext
⊕ = [email protected] (XOR) operation

Compare this with Equation (3.3) for the Vigenère cipher.
Thus, the ciphertext is generated by performing the bitwise XOR of the plain-

text and the key. Because of the properties of the XOR, decryption simply involves
the same bitwise operation:

pi = ci⊕ ki

which compares with Equation (3.4).
The essence of this technique is the means of construction of the key. Vernam

proposed the use of a running loop of tape that eventually repeated the key, so that
in fact the system worked with a very long but repeating keyword. Although such
a scheme, with a long key, presents formidable cryptanalytic difficulties, it can be
broken with sufficient ciphertext, the use of known or probable plaintext sequences,
or both.

One-Time Pad

An Army Signal Corp officer, Joseph Mauborgne, proposed an improvement to the
Vernam cipher that yields the ultimate in security. Mauborgne suggested using a
random key that is as long as the message, so that the key need not be repeated. In
addition, the key is to be used to encrypt and decrypt a single message, and then is
discarded. Each new message requires a new key of the same length as the new mes-
sage. Such a scheme, known as a one-time pad, is unbreakable. It produces random
output that bears no statistical relationship to the plaintext. Because the ciphertext

Figure 3.7 Vernam Cipher

Key stream
generator

Cryptographic
bit stream (ki)

Cryptographic
bit stream (ki)

Plaintext
(pi)

Plaintext
(pi)

Ciphertext
(ci )

Key stream
generator

106 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

contains no information whatsoever about the plaintext, there is simply no way to
break the code.

An example should illustrate our point. Suppose that we are using a Vigenère
scheme with 27 characters in which the twenty-seventh character is the space
character, but with a one-time key that is as long as the message. Consider the
ciphertext

ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS

We now show two different decryptions using two different keys:

ciphertext: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS
key: pxlmvmsydofuyrvzwc tnlebnecvgdupahfzzlmnyih
plaintext: mr mustard with the candlestick in the hall

ciphertext: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS
key: pftgpmiydgaxgoufhklllmhsqdqogtewbqfgyovuhwt
plaintext: miss scarlet with the knife in the library

Suppose that a cryptanalyst had managed to find these two keys. Two plau-
sible plaintexts are produced. How is the cryptanalyst to decide which is the correct
decryption (i.e., which is the correct key)? If the actual key were produced in a truly
random fashion, then the cryptanalyst cannot say that one of these two keys is more
likely than the other. Thus, there is no way to decide which key is correct and there-
fore which plaintext is correct.

In fact, given any plaintext of equal length to the ciphertext, there is a key that
produces that plaintext. Therefore, if you did an exhaustive search of all possible
keys, you would end up with many legible plaintexts, with no way of knowing which
was the intended plaintext. Therefore, the code is unbreakable.

The security of the one-time pad is entirely due to the randomness of the key.
If the stream of characters that constitute the key is truly random, then the stream
of characters that constitute the ciphertext will be truly random. Thus, there are no
patterns or regularities that a cryptanalyst can use to attack the ciphertext.

In theory, we need look no further for a cipher. The one-time pad offers com-
plete security but, in practice, has two fundamental difficulties:

1. There is the practical problem of making large quantities of random keys. Any
heavily used system might require millions of random characters on a regular
basis. Supplying truly random characters in this volume is a significant task.

2. Even more daunting is the problem of key distribution and protection. For
every message to be sent, a key of equal length is needed by both sender and
receiver. Thus, a mammoth key distribution problem exists.

Because of these difficulties, the one-time pad is of limited utility and is useful
primarily for low-bandwidth channels requiring very high security.

The one-time pad is the only cryptosystem that exhibits what is referred to as
perfect secrecy. This concept is explored in Appendix F.

3.3 / TRANSPOSITION TECHNIQUES 107

3.3 TRANSPOSITION TECHNIQUES

All the techniques examined so far involve the substitution of a ciphertext symbol
for a plaintext symbol. A very different kind of mapping is achieved by performing
some sort of permutation on the plaintext letters. This technique is referred to as a
transposition cipher.

The simplest such cipher is the rail fence technique, in which the plaintext is
written down as a sequence of diagonals and then read off as a sequence of rows.
For example, to encipher the message “meet me after the toga party” with a rail
fence of depth 2, we write the following:

m e m a t r h t g p r y
e t e f e t e o a a t

The encrypted message is

MEMATRHTGPRYETEFETEOAAT

This sort of thing would be trivial to cryptanalyze. A more complex scheme is
to write the message in a rectangle, row by row, and read the message off, column
by column, but permute the order of the columns. The order of the columns then
becomes the key to the algorithm. For example,

Key: 4 3 1 2 5 6 7
Plaintext: a t t a c k p
o s t p o n e
d u n t i l t
w o a m x y z
Ciphertext: TTNAAPTMTSUOAODWCOIXKNLYPETZ

Thus, in this example, the key is 4312567. To encrypt, start with the column
that is labeled 1, in this case column 3. Write down all the letters in that column.
Proceed to column 4, which is labeled 2, then column 2, then column 1, then
columns 5, 6, and 7.

A pure transposition cipher is easily recognized because it has the same letter
frequencies as the original plaintext. For the type of columnar transposition just
shown, cryptanalysis is fairly straightforward and involves laying out the cipher-
text in a matrix and playing around with column positions. Digram and trigram fre-
quency tables can be useful.

The transposition cipher can be made significantly more secure by perform-
ing more than one stage of transposition. The result is a more complex permutation
that is not easily reconstructed. Thus, if the foregoing message is reencrypted using
the same algorithm,

108 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

Key: 4 3 1 2 5 6 7
Input: t t n a a p t
m t s u o a o
d w c o i x k
n l y p e t z
Output: NSCYAUOPTTWLTMDNAOIEPAXTTOKZ

To visualize the result of this double transposition, designate the letters in the
original plaintext message by the numbers designating their position. Thus, with 28
letters in the message, the original sequence of letters is

01 02 03 04 05 06 07 08 09 10 11 12 13 14
15 16 17 18 19 20 21 22 23 24 25 26 27 28

After the first transposition, we have

03 10 17 24 04 11 18 25 02 09 16 23 01 08
15 22 05 12 19 26 06 13 20 27 07 14 21 28

which has a somewhat regular structure. But after the second transposition, we have

17 09 05 27 24 16 12 07 10 02 22 20 03 25
15 13 04 23 19 14 11 01 26 21 18 08 06 28

This is a much less structured permutation and is much more difficult to cryptanalyze.

3.4 ROTOR MACHINES

The example just given suggests that multiple stages of encryption can produce an
algorithm that is significantly more difficult to cryptanalyze. This is as true of substi-
tution ciphers as it is of transposition ciphers. Before the introduction of DES, the
most important application of the principle of multiple stages of encryption was a
class of systems known as rotor machines.10

The basic principle of the rotor machine is illustrated in Figure 3.8. The
machine consists of a set of independently rotating cylinders through which electri-
cal pulses can flow. Each cylinder has 26 input pins and 26 output pins, with internal
wiring that connects each input pin to a unique output pin. For simplicity, only three
of the internal connections in each cylinder are shown.

If we associate each input and output pin with a letter of the alphabet, then a
single cylinder defines a monoalphabetic substitution. For example, in Figure 3.8,
if an operator depresses the key for the letter A, an electric signal is applied to

10Machines based on the rotor principle were used by both Germany (Enigma) and Japan (Purple) in
World War II. The breaking of both codes by the Allies was a significant factor in the war’s outcome.

3.4 / ROTOR MACHINES 109

the first pin of the first cylinder and flows through the internal connection to the
twenty-fifth output pin.

Consider a machine with a single cylinder. After each input key is depressed,
the cylinder rotates one position, so that the internal connections are shifted accord-
ingly. Thus, a different monoalphabetic substitution cipher is defined. After 26 let-
ters of plaintext, the cylinder would be back to the initial position. Thus, we have a
polyalphabetic substitution algorithm with a period of 26.

A single-cylinder system is trivial and does not present a formidable crypt-
analytic task. The power of the rotor machine is in the use of multiple cylinders, in
which the output pins of one cylinder are connected to the input pins of the next.
Figure 3.8 shows a three-cylinder system. The left half of the figure shows a position
in which the input from the operator to the first pin (plaintext letter a) is routed
through the three cylinders to appear at the output of the second pin (ciphertext
letter B).

With multiple cylinders, the one closest to the operator input rotates one
pin position with each keystroke. The right half of Figure 3.8 shows the system’s
configuration after a single keystroke. For every complete rotation of the inner
cylinder, the middle cylinder rotates one pin position. Finally, for every complete
rotation of the middle cylinder, the outer cylinder rotates one pin position. This
is the same type of operation seen with an odometer. The result is that there are
26 * 26 * 26 = 17,576 different substitution alphabets used before the system

Figure 3.8 Three-Rotor Machine with Wiring Represented by Numbered Contacts

24
25
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Direction of motion Direction of motion

Fast rotor Medium rotor Slow rotor Fast rotor Medium rotor Slow rotor
(a) Initial setting (b) Setting after one keystroke

A
B
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Hiva-Network.Com

110 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

repeats. The addition of fourth and fifth rotors results in periods of 456,976 and
11,881,376 letters, respectively. Thus, a given setting of a 5-rotor machine is equiva-
lent to a Vigenère cipher with a key length of 11,881,376.

Such a scheme presents a formidable cryptanalytic challenge. If, for example,
the cryptanalyst attempts to use a letter frequency analysis approach, the analyst
is faced with the equivalent of over 11 million monoalphabetic ciphers. We might
need on the order of 50 letters in each monalphabetic cipher for a solution, which
means that the analyst would need to be in possession of a ciphertext with a length
of over half a billion letters.

The significance of the rotor machine today is that it points the way to a large
class of symmetric ciphers, of which the Data Encryption Standard (DES) is the
most prominent. DES is introduced in Chapter 4.

3.5 STEGANOGRAPHY

We conclude with a discussion of a technique that (strictly speaking), is not encryp-
tion, namely, steganography.

A plaintext message may be hidden in one of two ways. The methods of
steganography conceal the existence of the message, whereas the methods of cryp-
tography render the message unintelligible to outsiders by various transformations
of the text.11

A simple form of steganography, but one that is time-consuming to construct,
is one in which an arrangement of words or letters within an apparently innocuous
text spells out the real message. For example, the sequence of first letters of each
word of the overall message spells out the hidden message. Figure 3.9 shows an
example in which a subset of the words of the overall message is used to convey the
hidden message. See if you can decipher this; it’s not too hard.

Various other techniques have been used historically; some examples are the
following [MYER91]:

■ Character marking: Selected letters of printed or typewritten text are over-
written in pencil. The marks are ordinarily not visible unless the paper is held
at an angle to bright light.

■ Invisible ink: A number of substances can be used for writing but leave no vis-
ible trace until heat or some chemical is applied to the paper.

■ Pin punctures: Small pin punctures on selected letters are ordinarily not vis-
ible unless the paper is held up in front of a light.

■ Typewriter correction ribbon: Used between lines typed with a black ribbon,
the results of typing with the correction tape are visible only under a strong
light.

11Steganography was an obsolete word that was revived by David Kahn and given the meaning it has
today [KAHN96].

3.5 / STEGANOGRAPHY 111

Although these techniques may seem archaic, they have contemporary equiv-
alents. [WAYN09] proposes hiding a message by using the least significant bits of
frames on a CD. For example, the Kodak Photo CD format’s maximum resolution
is 3096 * 6144 pixels, with each pixel containing 24 bits of RGB color information.
The least significant bit of each 24-bit pixel can be changed without greatly affecting
the quality of the image. The result is that you can hide a 130-kB message in a single
digital snapshot. There are now a number of software packages available that take
this type of approach to steganography.

Steganography has a number of drawbacks when compared to encryption.
It requires a lot of overhead to hide a relatively few bits of information, although
using a scheme like that proposed in the preceding paragraph may make it more
effective. Also, once the system is discovered, it becomes virtually worthless. This
problem, too, can be overcome if the insertion method depends on some sort of key
(e.g., see Problem 3.22). Alternatively, a message can be first encrypted and then
hidden using steganography.

The advantage of steganography is that it can be employed by parties who
have something to lose should the fact of their secret communication (not necessar-
ily the content) be discovered. Encryption flags traffic as important or secret or may
identify the sender or receiver as someone with something to hide.

Figure 3.9 A Puzzle for Inspector Morse

(From The Silent World of Nicholas Quinn, by Colin Dexter)

112 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

3.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

block cipher
brute-force attack
Caesar cipher
cipher
ciphertext
computationally secure
conventional encryption
cryptanalysis
cryptographic system
cryptography

cryptology
deciphering
decryption
digram
enciphering
encryption
Hill cipher
monoalphabetic cipher
one-time pad
plaintext

Playfair cipher
polyalphabetic cipher
rail fence cipher
single-key encryption
steganography
stream cipher
symmetric encryption
transposition cipher
unconditionally secure
Vigenère cipher

Review Questions

3.1 Describe the main requirements for the secure use of symmetric encryption.
3.2 What are the two basic functions used in encryption algorithms?
3.3 Differentiate between secret-key encryption and public-key encryption.
3.4 What is the difference between a block cipher and a stream cipher?
3.5 What are the two general approaches to attacking a cipher?
3.6 List and briefly define types of cryptanalytic attacks based on what is known to the

attacker.
3.7 What is the difference between an unconditionally secure cipher and a computation-

ally secure cipher?
3.8 Why is the Caesar cipher substitution technique vulnerable to a brute-force cryptanalysis?
3.9 How much key space is available when a monoalphabetic substitution cipher is used

to replace plaintext with ciphertext?
3.10 What is the drawback of a Playfair cipher?
3.11 What is the difference between a monoalphabetic cipher and a polyalphabetic cipher?
3.12 What are two problems with the one-time pad?
3.13 What is a transposition cipher?
3.14 What are the drawbacks of Steganography?

Problems

3.1 A generalization of the Caesar cipher, known as the affine Caesar cipher, has the fol-
lowing form: For each plaintext letter p, substitute the ciphertext letter C:

C = E([a, b], p) = (ap + b) mod 26

A basic requirement of any encryption algorithm is that it be one-to-one. That is, if
p ≠ q, then E(k, p) ≠ E(k, q). Otherwise, decryption is impossible, because more
than one plaintext character maps into the same ciphertext character. The affine
Caesar cipher is not one-to-one for all values of a. For example, for a = 2 and b = 3,
then E([a, b], 0) = E([a, b], 13) = 3.

a. Are there any limitations on the value of b? Explain why or why not.
b. Determine which values of a are not allowed.

3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 113

c. Provide a general statement of which values of a are and are not allowed. Justify
your statement.

3.2 How many one-to-one affine Caesar ciphers are there?
3.3 A ciphertext has been generated with an affine cipher. The most frequent letter of

the ciphertext is “C,” and the second most frequent letter of the ciphertext is “Z.”
Break this code.

3.4 The following ciphertext was generated using a simple substitution algorithm.

hzsrnqc klyy wqc flo mflwf ol zqdn nsoznj wskn lj xzsrbjnf,
wzsxz gqv zqhhnf ol ozn glco zlfnco hnlhrn; nsoznj jnrqosdnc
lj fnqj kjsnfbc, wzsxz sc xnjoqsfrv gljn efeceqr. zn rsdnb
qrlfn sf zsc zlecn sf cqdsrrn jlw, wzsoznj flfn hnfnojqonb.
q csfyrn blgncosx cekksxnb ol cnjdn zsg. zn pjnqmkqconb qfb
bsfnb qo ozn xrep, qo zlejc gqozngqosxqrrv ksanb, sf ozn cqgn
jllg, qo ozn cqgn oqprn, fndnj oqmsfy zsc gnqrc wsoz loznj
gngpnjc, gexz rncc pjsfysfy q yenco wsoz zsg; qfb wnfo zlgn
qo naqxorv gsbfsyzo, lfrv ol jnosjn qo lfxn ol pnb. zn fndnj
ecnb ozn xlcv xzqgpnjc wzsxz ozn jnkljg hjldsbnc klj soc
kqdlejnb gngpnjc. zn hqccnb onf zlejc leo lk ozn ownfov-klej
sf cqdsrrn jlw, nsoznj sf crnnhsfy lj gqmsfy zsc olsrno.

Decrypt this message.
Hints:

1. As you know, the most frequently occurring letter in English is e. Therefore, the
first or second (or perhaps third?) most common character in the message is likely
to stand for e. Also, e is often seen in pairs (e.g., meet, fleet, speed, seen, been,
agree, etc.). Try to find a character in the ciphertext that decodes to e.

2. The most common word in English is “the.” Use this fact to guess the characters
that stand for t and h.

3. Decipher the rest of the message by deducing additional words.
Warning: The resulting message is in English but may not make much sense on a first

reading.
3.5 One way to solve the key distribution problem is to use a line from a book that both

the sender and the receiver possess. Typically, at least in spy novels, the first sentence
of a book serves as the key. The particular scheme discussed in this problem is from
one of the best suspense novels involving secret codes, Talking to Strange Men, by
Ruth Rendell. Work this problem without consulting that book!

Consider the following message:

SIDKHKDM AF HCRKIABIE SHIMC KD LFEAILA

This ciphertext was produced using the first sentence of The Other Side of Silence
(a book about the spy Kim Philby):

The snow lay thick on the steps and the snowflakes driven by the wind
looked black in the headlights of the cars.

A simple substitution cipher was used.
a. What is the encryption algorithm?
b. How secure is it?
c. To make the key distribution problem simple, both parties can agree to use the first or

last sentence of a book as the key. To change the key, they simply need to agree on a
new book. The use of the first sentence would be preferable to the use of the last. Why?

3.6 In one of his cases, Sherlock Holmes was confronted with the following message.

534 C2 13 127 36 31 4 17 21 41
DOUGLAS 109 293 5 37 BIRLSTONE

26 BIRLSTONE 9 127 171

114 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

Although Watson was puzzled, Holmes was able immediately to deduce the type of
cipher. Can you?

3.7 This problem uses a real-world example, from an old U.S. Special Forces manual
(public domain). The document, filename SpecialForces.pdf, is available at box.com/
Crypto7e.
a. Using the two keys (memory words) cryptographic and network security, encrypt

the following message:

Be at the third pillar from the left outside the lyceum theatre tonight at seven.
If you are distrustful bring two friends.

Make reasonable assumptions about how to treat redundant letters and excess
letters in the memory words and how to treat spaces and punctuation. Indicate
what your assumptions are. Note: The message is from the Sherlock Holmes novel,
The Sign of Four.

b. Decrypt the ciphertext. Show your work.
c. Comment on when it would be appropriate to use this technique and what its

advantages are.

3.8 A disadvantage of the general monoalphabetic cipher is that both sender and receiver
must commit the permuted cipher sequence to memory. A common technique for
avoiding this is to use a keyword from which the cipher sequence can be gener-
ated. For example, using the keyword CRYPTO, write out the keyword followed by
unused letters in normal order and match this against the plaintext letters:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y z

cipher: C R Y P T O A B D E F G H I J K L M N Q S U V W X Z

If it is felt that this process does not produce sufficient mixing, write the remain-
ing letters on successive lines and then generate the sequence by reading down the
columns:

C R Y P T O

A B D E F G

H I J K L M

N Q S U V W

X Z

This yields the sequence:

C A H N X R B I Q Z Y D J S P E K U T F L V O G M W

Such a system is used in the example in Section 3.2 (the one that begins “it was
disclosed yesterday”). Determine the keyword.

3.9 When the PT-109 American patrol boat, under the command of Lieutenant John F.
Kennedy, was sunk by a Japanese destroyer, a message was received at an Australian
wireless station in Playfair code:

KXJEY UREBE ZWEHE WRYTU HEYFS

KREHE GOYFI WTTTU OLKSY CAJPO

BOTEI ZONTX BYBNT GONEY CUZWR

GDSON SXBOU YWRHE BAAHY USEDQ

The key used was royal new zealand navy. Decrypt the message. Translate TT into tt.

3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 115

3.10 a. Construct a Playfair matrix with the key algorithm.
b. Construct a Playfair matrix with the key cryptography. Make a reasonable assump-

tion about how to treat redundant letters in the key.
3.11 a. Using this Playfair matrix:

J/K C D E F

U N P Q S

Z V W X Y

R A L G O

B I T H M

Encrypt this message:

I only regret that I have but one life to give for my country.

Note: This message is by Nathan Hale, a soldier in the American Revolutionary War.
b. Repeat part (a) using the Playfair matrix from Problem 3.10a.
c. How do you account for the results of this problem? Can you generalize your

conclusion?
3.12 a. How many possible keys does the Playfair cipher have? Ignore the fact that

some keys might produce identical encryption results. Express your answer as an
approximate power of 2.

b. Now take into account the fact that some Playfair keys produce the same encryp-
tion results. How many effectively unique keys does the Playfair cipher have?

3.13 What substitution system results when we use a 1 * 25 Playfair matrix?
3.14 a. Encrypt the message “meet me at the usual place at ten rather than eight o clock”

using the Hill cipher with the key ¢7 3
2 5

≤. Show your calculations and the result.
b. Show the calculations for the corresponding decryption of the ciphertext to

recover the original plaintext.
3.15 We have shown that the Hill cipher succumbs to a known plaintext attack if sufficient

plaintext–ciphertext pairs are provided. It is even easier to solve the Hill cipher if a
chosen plaintext attack can be mounted. Describe such an attack.

3.16 It can be shown that the Hill cipher with the matrix ¢a b
c d

≤ requires that (ad - bc)
is relatively prime to 26; that is, the only common positive integer factor of (ad - bc)
and 26 is 1. Thus, if (ad - bc) = 13 or is even, the matrix is not allowed. Determine
the number of different (good) keys there are for a 2 * 2 Hill cipher without count-
ing them one by one, using the following steps:
a. Find the number of matrices whose determinant is even because one or both rows

are even. (A row is “even” if both entries in the row are even.)
b. Find the number of matrices whose determinant is even because one or both col-

umns are even. (A column is “even” if both entries in the column are even.)
c. Find the number of matrices whose determinant is even because all of the entries

are odd.
d. Taking into account overlaps, find the total number of matrices whose determi-

nant is even.
e. Find the number of matrices whose determinant is a multiple of 13 because the

first column is a multiple of 13.

116 CHAPTER 3 / CLASSICAL ENCRYPTION TECHNIQUES

f. Find the number of matrices whose determinant is a multiple of 13 where
the first column is not a multiple of 13 but the second column is a mul-
tiple of the first modulo 13.

g. Find the total number of matrices whose determinant is a multiple of 13.
h. Find the number of matrices whose determinant is a multiple of 26

because they fit cases parts (a) and (e), (b) and (e), (c) and (e), (a) and
(f), and so on.

i. Find the total number of matrices whose determinant is neither a mul-
tiple of 2 nor a multiple of 13.

3.17 Calculate the determinant mod 26 of

a. ¢2 3 5
1 3 7

≤ b. £2 1 1 3 2 55 7 1 8
3 1 4 1 2

≥
3.18 Determine the inverse mod 26 of

a. ¢2 3
1 22

≤ b. £ 6 24 113 16 10
20 17 15

≥
3.19 Using the Vigenère cipher, encrypt the word “cryptographic” using the word

“eng”.
3.20 This problem explores the use of a one-time pad version of the Vigenère

cipher. In this scheme, the key is a stream of random numbers between 0
and 26. For example, if the key is 3 19 5 . . . , then the first letter of plaintext
is encrypted with a shift of 3 letters, the second with a shift of 19 letters, the
third with a shift of 5 letters, and so on.
a. Encrypt the plaintext sendmoremoney with the key stream

3 11 5 7 17 21 0 11 14 8 7 13 9

b. Using the ciphertext produced in part (a), find a key so that the cipher-
text decrypts to the plaintext cashnotneeded.

3.21 What is the message embedded in Figure 3.9?
3.22 In one of Dorothy Sayers’s mysteries, Lord Peter is confronted with the

message shown in Figure 3.10. He also discovers the key to the message,
which is a sequence of integers:

787656543432112343456567878878765654

3432112343456567878878765654433211234

a. Decrypt the message. Hint: What is the largest integer value?
b. If the algorithm is known but not the key, how secure is the scheme?
c. If the key is known but not the algorithm, how secure is the scheme?

Figure 3.10 A Puzzle for Lord Peter

I thought to see the fairies in the fields, but I saw only the evil elephants with their black
backs. Woe! how that sight awed me! The elves danced all around and about while I heard
voices calling clearly. Ah! how I tried to see—throw off the ugly cloud—but no blind eye
of a mortal was permitted to spy them. So then came minstrels, having gold trumpets, harps
and drums. These played very loudly beside me, breaking that spell. So the dream vanished,
whereat I thanked Heaven. I shed many tears before the thin moon rose up, frail and faint as
a sickle of straw. Now though the Enchanter gnash his teeth vainly, yet shall he return as the
Spring returns. Oh, wretched man! Hell gapes, Erebus now lies open. The mouths of Death
wait on thy end.

3.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 117

Programming Problems

3.23 Write a program that can encrypt and decrypt using the general Caesar
cipher, also known as an additive cipher.

3.24 Write a program that can encrypt and decrypt using the affine cipher
described in Problem 3.1.

3.25 Write a program that can perform a letter frequency attack on an additive
cipher without human intervention. Your software should produce possible
plaintexts in rough order of likelihood. It would be good if your user inter-
face allowed the user to specify “give me the top 10 possible plaintexts.”

3.26 Write a program that can perform a letter frequency attack on any mono-
alphabetic substitution cipher without human intervention. Your software
should produce possible plaintexts in rough order of likelihood. It would
be good if your user interface allowed the user to specify “give me the top
10 possible plaintexts.”

3.27 Create software that can encrypt and decrypt using a 2 * 2 Hill cipher.
3.28 Create software that can perform a fast known plaintext attack on a Hill cipher,

given the dimension m. How fast are your algorithms, as a function of m?

118118

4.1 Traditional Block Cipher Structure

Stream Ciphers and Block Ciphers
Motivation for the Feistel Cipher Structure
The Feistel Cipher

4.2 The Data Encryption Standard

DES Encryption
DES Decryption

4.3 A DES Example

Results
The Avalanche Effect

4.4 The Strength of DES

The Use of 56-Bit Keys
The Nature of the DES Algorithm
Timing Attacks

4.5 Block Cipher Design Principles

Number of Rounds
Design of Function F
Key Schedule Algorithm

4.6 Key Terms, Review Questions, and Problems

CHAPTER

Block Ciphers and the Data
Encryption Standard

Hiva-Network.Com

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 119

The objective of this chapter is to illustrate the principles of modern symmetric
ciphers. For this purpose, we focus on the most widely used symmetric cipher: the Data
Encryption Standard (DES). Although numerous symmetric ciphers have been devel-
oped since the introduction of DES, and although it is destined to be replaced by the
Advanced Encryption Standard (AES), DES remains the most important such algo-
rithm. Furthermore, a detailed study of DES provides an understanding of the prin-
ciples used in other symmetric ciphers.

This chapter begins with a discussion of the general principles of symmetric block
ciphers, which are the principal type of symmetric ciphers studied in this book. The
other form of symmetric ciphers, stream ciphers, are discussed in Chapter 8. Next, we
cover full DES. Following this look at a specific algorithm, we return to a more general
discussion of block cipher design.

Compared to public-key ciphers, such as RSA, the structure of DES and most
symmetric ciphers is very complex and cannot be explained as easily as RSA and simi-
lar algorithms. Accordingly, the reader may wish to begin with a simplified version of
DES, which is described in Appendix G. This version allows the reader to perform
encryption and decryption by hand and gain a good understanding of the working of
the algorithm details. Classroom experience indicates that a study of this simplified
version enhances understanding of DES.1

4.1 TRADITIONAL BLOCK CIPHER STRUCTURE

Several important symmetric block encryption algorithms in current use are based
on a structure referred to as a Feistel block cipher [FEIS73]. For that reason, it is
important to examine the design principles of the Feistel cipher. We begin with a
comparison of stream ciphers and block ciphers. Then we discuss the motivation for
the Feistel block cipher structure. Finally, we discuss some of its implications.

1However, you may safely skip Appendix G, at least on a first reading. If you get lost or bogged down in
the details of DES, then you can go back and start with simplified DES.

LEARNING OBJECTIVES

After studying this chapter, you should be able to

◆ Understand the distinction between stream ciphers and block ciphers.

◆ Present an overview of the Feistel cipher and explain how decryption is
the inverse of encryption.

◆ Present an overview of Data Encryption Standard (DES).

◆ Explain the concept of the avalanche effect.

◆ Discuss the cryptographic strength of DES.

◆ Summarize the principal block cipher design principles.

120 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

Stream Ciphers and Block Ciphers

A stream cipher is one that encrypts a digital data stream one bit or one byte at a
time. Examples of classical stream ciphers are the autokeyed Vigenère cipher and
the Vernam cipher. In the ideal case, a one-time pad version of the Vernam cipher
would be used (Figure 3.7), in which the keystream (ki) is as long as the plaintext bit
stream (pi). If the cryptographic keystream is random, then this cipher is unbreakable
by any means other than acquiring the keystream. However, the keystream must be
provided to both users in advance via some independent and secure channel. This
introduces insurmountable logistical problems if the intended data traffic is very large.

Accordingly, for practical reasons, the bit-stream generator must be imple-
mented as an algorithmic procedure, so that the cryptographic bit stream can be
produced by both users. In this approach (Figure 4.1a), the bit-stream generator is
a key-controlled algorithm and must produce a bit stream that is cryptographically
strong. That is, it must be computationally impractical to predict future portions of
the bit stream based on previous portions of the bit stream. The two users need only
share the generating key, and each can produce the keystream.

A block cipher is one in which a block of plaintext is treated as a whole and
used to produce a ciphertext block of equal length. Typically, a block size of 64 or

Figure 4.1 Stream Cipher and Block Cipher

Bit-stream
generation
algorithm

ENCRYPTION

(a) Stream cipher using algorithmic bit-stream generator

(b) Block cipher

Key
( K )

Encryption
algorithm

Plaintext

b bits

b bits

Key
( K )

ki

Plaintext
(pi)

Plaintext
(pi)

Bit-stream
generation
algorithm

DECRYPTION

Key
( K )

ki

Ciphertext
(ci)

Ciphertext

Decryption
algorithm

Ciphertext

b bits

b bits

Key
( K )

Plaintext

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 121

128 bits is used. As with a stream cipher, the two users share a symmetric encryption
key (Figure 4.1b). Using some of the modes of operation explained in Chapter 7, a
block cipher can be used to achieve the same effect as a stream cipher.

Far more effort has gone into analyzing block ciphers. In general, they seem
applicable to a broader range of applications than stream ciphers. The vast majority
of network-based symmetric cryptographic applications make use of block ciphers.
Accordingly, the concern in this chapter, and in our discussions throughout the
book of symmetric encryption, will primarily focus on block ciphers.

Motivation for the Feistel Cipher Structure

A block cipher operates on a plaintext block of n bits to produce a ciphertext block
of n bits. There are 2n possible different plaintext blocks and, for the encryption
to be reversible (i.e., for decryption to be possible), each must produce a unique
ciphertext block. Such a transformation is called reversible, or nonsingular. The fol-
lowing examples illustrate nonsingular and singular transformations for n = 2.

Reversible Mapping Irreversible Mapping

Plaintext Ciphertext Plaintext Ciphertext

00 11 00 11

01 10 01 10

10 00 10 01

11 01 11 01

In the latter case, a ciphertext of 01 could have been produced by one of two plain-
text blocks. So if we limit ourselves to reversible mappings, the number of different
transformations is 2n!.2

Figure 4.2 illustrates the logic of a general substitution cipher for n = 4.
A 4-bit input produces one of 16 possible input states, which is mapped by the sub-
stitution cipher into a unique one of 16 possible output states, each of which is repre-
sented by 4 ciphertext bits. The encryption and decryption mappings can be defined
by a tabulation, as shown in Table 4.1. This is the most general form of block cipher
and can be used to define any reversible mapping between plaintext and ciphertext.
Feistel refers to this as the ideal block cipher, because it allows for the maximum
number of possible encryption mappings from the plaintext block [FEIS75].

But there is a practical problem with the ideal block cipher. If a small block
size, such as n = 4, is used, then the system is equivalent to a classical substitution
cipher. Such systems, as we have seen, are vulnerable to a statistical analysis of the
plaintext. This weakness is not inherent in the use of a substitution cipher but rather
results from the use of a small block size. If n is sufficiently large and an arbitrary
reversible substitution between plaintext and ciphertext is allowed, then the statisti-
cal characteristics of the source plaintext are masked to such an extent that this type
of cryptanalysis is infeasible.

2The reasoning is as follows: For the first plaintext, we can choose any of 2n ciphertext blocks. For the
second plaintext, we choose from among 2n - 1 remaining ciphertext blocks, and so on.

122 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

An arbitrary reversible substitution cipher (the ideal block cipher) for a large
block size is not practical, however, from an implementation and performance
point of view. For such a transformation, the mapping itself constitutes the key.
Consider again Table 4.1, which defines one particular reversible mapping from

Figure 4.2 General n-bit-n-bit Block Substitution (shown with n = 4)

4-bit input

4 to 16 decoder

16 to 4 encoder

4-bit output

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Table 4.1 Encryption and Decryption Tables for Substitution Cipher of Figure 4.2

Plaintext Ciphertext

0000 1110

0001 0100

0010 1101

0011 0001

0100 0010

0101 1111

0110 1011

0111 1000

1000 0011

1001 1010

1010 0110

1011 1100

1100 0101

1101 1001

1110 0000

1111 0111

Ciphertext Plaintext

0000 1110

0001 0011

0010 0100

0011 1000

0100 0001

0101 1100

0110 1010

0111 1111

1000 0111

1001 1101

1010 1001

1011 0110

1100 1011

1101 0010

1110 0000

1111 0101

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 123

plaintext to ciphertext for n = 4. The mapping can be defined by the entries in the
second column, which show the value of the ciphertext for each plaintext block.
This, in essence, is the key that determines the specific mapping from among all
possible mappings. In this case, using this straightforward method of defining the
key, the required key length is (4 bits) * (16 rows) = 64 bits. In general, for an
n-bit ideal block cipher, the length of the key defined in this fashion is n * 2n bits.
For a 64-bit block, which is a desirable length to thwart statistical attacks, the
required key length is 64 * 264 = 270 ≈ 1021 bits.

In considering these difficulties, Feistel points out that what is needed is an
approximation to the ideal block cipher system for large n, built up out of compo-
nents that are easily realizable [FEIS75]. But before turning to Feistel’s approach,
let us make one other observation. We could use the general block substitution
cipher but, to make its implementation tractable, confine ourselves to a subset of
the 2n! possible reversible mappings. For example, suppose we define the mapping
in terms of a set of linear equations. In the case of n = 4, we have

y1 = k11x1 + k12x2 + k13x3 + k14x4
y2 = k21x1 + k22x2 + k23x3 + k24x4
y3 = k31x1 + k32x2 + k33x3 + k34x4
y4 = k41x1 + k42x2 + k43x3 + k44x4

where the xi are the four binary digits of the plaintext block, the yi are the four bi-
nary digits of the ciphertext block, the kij are the binary coefficients, and arithmetic
is mod 2. The key size is just n2, in this case 16 bits. The danger with this kind of for-
mulation is that it may be vulnerable to cryptanalysis by an attacker that is aware of
the structure of the algorithm. In this example, what we have is essentially the Hill
cipher discussed in Chapter 3, applied to binary data rather than characters. As we
saw in Chapter 3, a simple linear system such as this is quite vulnerable.

The Feistel Cipher

Feistel proposed [FEIS73] that we can approximate the ideal block cipher by utiliz-
ing the concept of a product cipher, which is the execution of two or more simple
ciphers in sequence in such a way that the final result or product is cryptographi-
cally stronger than any of the component ciphers. The essence of the approach is
to develop a block cipher with a key length of k bits and a block length of n bits,
allowing a total of 2k possible transformations, rather than the 2n! transformations
available with the ideal block cipher.

In particular, Feistel proposed the use of a cipher that alternates substitutions
and permutations, where these terms are defined as follows:

■ Substitution: Each plaintext element or group of elements is uniquely replaced
by a corresponding ciphertext element or group of elements.

■ Permutation: A sequence of plaintext elements is replaced by a permutation
of that sequence. That is, no elements are added or deleted or replaced in the
sequence, rather the order in which the elements appear in the sequence is
changed.

124 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

In fact, Feistel’s is a practical application of a proposal by Claude Shannon
to develop a product cipher that alternates confusion and diffusion functions
[SHAN49].3 We look next at these concepts of diffusion and confusion and then
present the Feistel cipher. But first, it is worth commenting on this remarkable fact:
The Feistel cipher structure, which dates back over a quarter century and which, in
turn, is based on Shannon’s proposal of 1945, is the structure used by a number of
significant symmetric block ciphers currently in use. In particular, the Feistel struc-
ture is used for Triple Data Encryption Algorithm (TDEA), which is one of the two
encryption algorithms (along with AES), approved for general use by the National
Institute of Standards and Technology (NIST). The Feistel structure is also used for
several schemes for format-preserving encryption, which have recently come into
prominence. In addition, the Camellia block cipher is a Feistel structure; it is one
of the possible symmetric ciphers in TLS and a number of other Internet security
protocols. Both TDEA and format-preserving encryption are covered in Chapter 7.

DIFFUSION AND CONFUSION The terms diffusion and confusion were introduced by
Claude Shannon to capture the two basic building blocks for any cryptographic sys-
tem [SHAN49]. Shannon’s concern was to thwart cryptanalysis based on statisti-
cal analysis. The reasoning is as follows. Assume the attacker has some knowledge
of the statistical characteristics of the plaintext. For example, in a human-readable
message in some language, the frequency distribution of the various letters may be
known. Or there may be words or phrases likely to appear in the message (probable
words). If these statistics are in any way reflected in the ciphertext, the cryptanalyst
may be able to deduce the encryption key, part of the key, or at least a set of keys
likely to contain the exact key. In what Shannon refers to as a strongly ideal cipher,
all statistics of the ciphertext are independent of the particular key used. The arbi-
trary substitution cipher that we discussed previously (Figure 4.2) is such a cipher,
but as we have seen, it is impractical.4

Other than recourse to ideal systems, Shannon suggests two methods for
frustrating statistical cryptanalysis: diffusion and confusion. In diffusion, the sta-
tistical structure of the plaintext is dissipated into long-range statistics of the
ciphertext. This is achieved by having each plaintext digit affect the value of many
ciphertext digits; generally, this is equivalent to having each ciphertext digit be
affected by many plaintext digits. An example of diffusion is to encrypt a message
M = m1, m2, m3, c of characters with an averaging operation:

yn = ¢ ak
i=1

mn + i≤ mod 26
3The paper is available at box.com/Crypto7e. Shannon’s 1949 paper appeared originally as a classified
report in 1945. Shannon enjoys an amazing and unique position in the history of computer and informa-
tion science. He not only developed the seminal ideas of modern cryptography but is also responsible for
inventing the discipline of information theory. Based on his work in information theory, he developed
a formula for the capacity of a data communications channel, which is still used today. In addition, he
founded another discipline, the application of Boolean algebra to the study of digital circuits; this last he
managed to toss off as a master’s thesis.
4Appendix F expands on Shannon’s concepts concerning measures of secrecy and the security of crypto-
graphic algorithms.

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 125

adding k successive letters to get a ciphertext letter yn. One can show that the sta-
tistical structure of the plaintext has been dissipated. Thus, the letter frequencies in
the ciphertext will be more nearly equal than in the plaintext; the digram frequen-
cies will also be more nearly equal, and so on. In a binary block cipher, diffusion can
be achieved by repeatedly performing some permutation on the data followed by
applying a function to that permutation; the effect is that bits from different posi-
tions in the original plaintext contribute to a single bit of ciphertext.5

Every block cipher involves a transformation of a block of plaintext into a
block of ciphertext, where the transformation depends on the key. The mechanism
of diffusion seeks to make the statistical relationship between the plaintext and
ciphertext as complex as possible in order to thwart attempts to deduce the key. On
the other hand, confusion seeks to make the relationship between the statistics of
the ciphertext and the value of the encryption key as complex as possible, again to
thwart attempts to discover the key. Thus, even if the attacker can get some handle
on the statistics of the ciphertext, the way in which the key was used to produce that
ciphertext is so complex as to make it difficult to deduce the key. This is achieved by
the use of a complex substitution algorithm. In contrast, a simple linear substitution
function would add little confusion.

As [ROBS95b] points out, so successful are diffusion and confusion in captur-
ing the essence of the desired attributes of a block cipher that they have become the
cornerstone of modern block cipher design.

FEISTEL CIPHER STRUCTURE The left-hand side of Figure 4.3 depicts the encryption
structure proposed by Feistel. The inputs to the encryption algorithm are a plaintext
block of length 2w bits and a key K. The plaintext block is divided into two halves,
LE0 and RE0. The two halves of the data pass through n rounds of processing and
then combine to produce the ciphertext block. Each round i has as inputs LEi- 1 and
REi- 1 derived from the previous round, as well as a subkey Ki derived from the over-
all K. In general, the subkeys Ki are different from K and from each other. In Figure
4.3, 16 rounds are used, although any number of rounds could be implemented.

All rounds have the same structure. A substitution is performed on the left
half of the data. This is done by applying a round function F to the right half of the
data and then taking the exclusive-OR of the output of that function and the left
half of the data. The round function has the same general structure for each round
but is parameterized by the round subkey Ki. Another way to express this is to say
that F is a function of right-half block of w bits and a subkey of y bits, which pro-
duces an output value of length w bits: F(REi, Ki+ 1). Following this substitution, a
permutation is performed that consists of the interchange of the two halves of the
data.6 This structure is a particular form of the substitution-permutation network
(SPN) proposed by Shannon.

5Some books on cryptography equate permutation with diffusion. This is incorrect. Permutation, by itself,
does not change the statistics of the plaintext at the level of individual letters or permuted blocks. For exam-
ple, in DES, the permutation swaps two 32-bit blocks, so statistics of strings of 32 bits or less are preserved.
6The final round is followed by an interchange that undoes the interchange that is part of the final round.
One could simply leave both interchanges out of the diagram, at the sacrifice of some consistency of pre-
sentation. In any case, the effective lack of a swap in the final round is done to simplify the implementa-
tion of the decryption process, as we shall see.

126 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

The exact realization of a Feistel network depends on the choice of the follow-
ing parameters and design features:

■ Block size: Larger block sizes mean greater security (all other things being
equal) but reduced encryption/decryption speed for a given algorithm. The
greater security is achieved by greater diffusion. Traditionally, a block size of
64 bits has been considered a reasonable tradeoff and was nearly universal in
block cipher design. However, the new AES uses a 128-bit block size.

Figure 4.3 Feistel Encryption and Decryption (16 rounds)

Output (ciphertext)

K1

LD0 = RE16 RD0 = LE16

LD2 = RE14 RD2 = LE14

LD14 = RE2 RD14 = LE2

LD16 = RE0

LD17 = RE0

RD16 = LE0

RD17 = LE0

RD1 = LE15LD1 = RE15

RD15 = LE1LD15 = RE1

Input (ciphertext)

Output (plaintext)

R
ou

nd
1

K1

K2

K15

K16

K2

K15

K16

F

LE0 RE0

Input (plaintext)

LE1 RE1

LE2 RE2

F

F

LE14 RE14

LE15 RE15

LE16 RE16

LE17 RE17

F

F

F

F

F

R
ou

nd
2

R
ou

nd
1

5
R

ou
nd

1
6

R
ou

nd
1

6
R

ou
nd

1
5

R
ou

nd
2

R
ou

nd
1

4.1 / TRADITIONAL BLOCK CIPHER STRUCTURE 127

■ Key size: Larger key size means greater security but may decrease encryption/
decryption speed. The greater security is achieved by greater resistance to
brute-force attacks and greater confusion. Key sizes of 64 bits or less are now
widely considered to be inadequate, and 128 bits has become a common size.

■ Number of rounds: The essence of the Feistel cipher is that a single round
offers inadequate security but that multiple rounds offer increasing security.
A typical size is 16 rounds.

■ Subkey generation algorithm: Greater complexity in this algorithm should
lead to greater difficulty of cryptanalysis.

■ Round function F: Again, greater complexity generally means greater resis-
tance to cryptanalysis.

There are two other considerations in the design of a Feistel cipher:

■ Fast software encryption/decryption: In many cases, encryption is embedded
in applications or utility functions in such a way as to preclude a hardware im-
plementation. Accordingly, the speed of execution of the algorithm becomes a
concern.

■ Ease of analysis: Although we would like to make our algorithm as difficult as
possible to cryptanalyze, there is great benefit in making the algorithm easy
to analyze. That is, if the algorithm can be concisely and clearly explained, it is
easier to analyze that algorithm for cryptanalytic vulnerabilities and therefore
develop a higher level of assurance as to its strength. DES, for example, does
not have an easily analyzed functionality.

FEISTEL DECRYPTION ALGORITHM The process of decryption with a Feistel cipher
is essentially the same as the encryption process. The rule is as follows: Use the
ciphertext as input to the algorithm, but use the subkeys Ki in reverse order. That
is, use Kn in the first round, Kn - 1 in the second round, and so on, until K1 is used in
the last round. This is a nice feature, because it means we need not implement two
different algorithms; one for encryption and one for decryption.

To see that the same algorithm with a reversed key order produces the cor-
rect result, Figure 4.3 shows the encryption process going down the left-hand side
and the decryption process going up the right-hand side for a 16-round algorithm.
For clarity, we use the notation LEi and REi for data traveling through the encryp-
tion algorithm and LDi and RDi for data traveling through the decryption algo-
rithm. The diagram indicates that, at every round, the intermediate value of the
decryption process is equal to the corresponding value of the encryption process
with the two halves of the value swapped. To put this another way, let the output
of the ith encryption round be LEi ‘REi (LEi concatenated with REi). Then the cor-
responding output of the (16 - i)th decryption round is REi ‘LEi or, equivalently,
LD16 - i ‘RD16 - i.

Let us walk through Figure 4.3 to demonstrate the validity of the preceding
assertions. After the last iteration of the encryption process, the two halves of the
output are swapped, so that the ciphertext is RE16 ‘LE16. The output of that round
is the ciphertext. Now take that ciphertext and use it as input to the same algorithm.
The input to the first round is RE16 ‘LE16, which is equal to the 32-bit swap of the
output of the sixteenth round of the encryption process.

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128 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

Now we would like to show that the output of the first round of the decryption
process is equal to a 32-bit swap of the input to the sixteenth round of the encryp-
tion process. First, consider the encryption process. We see that

LE16 = RE15
RE16 = LE15⊕ F(RE15, K16)

On the decryption side,

LD1 = RD0 = LE16 = RE15
RD1 = LD0⊕ F(RD0, K16)

= RE16⊕ F(RE15, K16)
= [LE15⊕ F(RE15, K16)]⊕ F(RE15, K16)

The XOR has the following properties:

[A⊕ B]⊕ C = A⊕ [B⊕ C]
D⊕D = 0
E⊕ 0 = E

Thus, we have LD1 = RE15 and RD1 = LE15. Therefore, the output of the first
round of the decryption process is RE15 ‘LE15, which is the 32-bit swap of the input
to the sixteenth round of the encryption. This correspondence holds all the way
through the 16 iterations, as is easily shown. We can cast this process in general
terms. For the ith iteration of the encryption algorithm,

LEi = REi- 1
REi = LEi- 1⊕ F(REi- 1, Ki)

Rearranging terms:

REi- 1 = LEi
LEi- 1 = REi⊕ F(REi- 1, Ki) = REi⊕ F(LEi, Ki)

Thus, we have described the inputs to the ith iteration as a function of the outputs, and
these equations confirm the assignments shown in the right-hand side of Figure 4.3.

Finally, we see that the output of the last round of the decryption process is
RE0 ‘LE0. A 32-bit swap recovers the original plaintext, demonstrating the validity
of the Feistel decryption process.

Note that the derivation does not require that F be a reversible function. To
see this, take a limiting case in which F produces a constant output (e.g., all ones)
regardless of the values of its two arguments. The equations still hold.

To help clarify the preceding concepts, let us look at a specific example
(Figure 4.4 and focus on the fifteenth round of encryption, corresponding to the sec-
ond round of decryption. Suppose that the blocks at each stage are 32 bits (two 16-bit
halves) and that the key size is 24 bits. Suppose that at the end of encryption round
fourteen, the value of the intermediate block (in hexadecimal) is DE7F03A6. Then
LE14 = DE7F and RE14 = 03A6. Also assume that the value of K15 is 12DE52.
After round 15, we have LE15 = 03A6 and RE15 = F(03A6, 12DE52)⊕DE7F.

4.2 / THE DATA ENCRYPTION STANDARD 129

Now let’s look at the decryption. We assume that LD1 = RE15 and
RD1 = LE15, as shown in Figure 4.3, and we want to demonstrate that LD2 = RE14
and RD2 = LE14. So, we start with LD1 = F(03A6, 12DE52)⊕DE7F and
RD1 = 03A6. Then, from Figure 4.3, LD2 = 03A6 = RE14 and RD2 =
F(03A6, 12DE52)⊕ [F(03A6, 12DE52)⊕DE7F] = DE7F = LE14.

4.2 THE DATA ENCRYPTION STANDARD

Until the introduction of the Advanced Encryption Standard (AES) in 2001, the
Data Encryption Standard (DES) was the most widely used encryption scheme.
DES was issued in 1977 by the National Bureau of Standards, now the National
Institute of Standards and Technology (NIST), as Federal Information Processing
Standard 46 (FIPS PUB 46). The algorithm itself is referred to as the Data
Encryption Algorithm (DEA).7 For DEA, data are encrypted in 64-bit blocks using
a 56-bit key. The algorithm transforms 64-bit input in a series of steps into a 64-bit
output. The same steps, with the same key, are used to reverse the encryption.

Over the years, DES became the dominant symmetric encryption algorithm,
especially in financial applications. In 1994, NIST reaffirmed DES for federal use
for another five years; NIST recommended the use of DES for applications other
than the protection of classified information. In 1999, NIST issued a new version
of its standard (FIPS PUB 46-3) that indicated that DES should be used only
for legacy systems and that triple DES (which in essence involves repeating the
DES algorithm three times on the plaintext using two or three different keys to
produce the ciphertext) be used. We study triple DES in Chapter 7. Because the
underlying encryption and decryption algorithms are the same for DES and triple
DES, it remains important to understand the DES cipher. This section provides an
overview.For the interested reader, Appendix S provides further detail.

7The terminology is a bit confusing. Until recently, the terms DES and DEA could be used interchange-
ably. However, the most recent edition of the DES document includes a specification of the DEA
described here plus the triple DEA (TDEA) described in Chapter 7. Both DEA and TDEA are part of
the Data Encryption Standard. Further, until the recent adoption of the official term TDEA, the triple
DEA algorithm was typically referred to as triple DES and written as 3DES. For the sake of convenience,
we will use the term 3DES.

Figure 4.4 Feistel Example

12DE52

12DE52

F

DE7F 03A6

Decryption roundEncryption round

03A6

6A306A30 F(03A6, 12DE52) DE7F F(03A6, 12DE52) DE7F

F(03A6, 12DE52)
[F(03A6, 12DE52) DE7F]

= DE7F

FR
ou

nd
1

5

R
ou

nd
2

130 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

DES Encryption

The overall scheme for DES encryption is illustrated in Figure 4.5. As with any
encryption scheme, there are two inputs to the encryption function: the plaintext to
be encrypted and the key. In this case, the plaintext must be 64 bits in length and the
key is 56 bits in length.8

Looking at the left-hand side of the figure, we can see that the processing
of the plaintext proceeds in three phases. First, the 64-bit plaintext passes through
an initial permutation (IP) that rearranges the bits to produce the permuted input.

8Actually, the function expects a 64-bit key as input. However, only 56 of these bits are ever used; the
other 8 bits can be used as parity bits or simply set arbitrarily.

Figure 4.5 General Depiction of DES Encryption Algorithm

Initial permutation

Permuted choice 2Round 1

32-bit swap

Inverse initial
permutation

Permuted choice 1

Round 2

Round 16

64-bit plaintext 64-bit key

K1

K2

K16

64-bit ciphertext

Left circular shift

Permuted choice 2 Left circular shift

Permuted choice 2 Left circular shift

64 56

56

56

56

48

48

48

56 64

64 bits

4.3 / A DES EXAMPLE 131

This is followed by a phase consisting of sixteen rounds of the same function, which
involves both permutation and substitution functions. The output of the last (six-
teenth) round consists of 64 bits that are a function of the input plaintext and the
key. The left and right halves of the output are swapped to produce the preoutput.
Finally, the preoutput is passed through a permutation [IP-1] that is the inverse of
the initial permutation function, to produce the 64-bit ciphertext. With the excep-
tion of the initial and final permutations, DES has the exact structure of a Feistel
cipher, as shown in Figure 4.3.

The right-hand portion of Figure 4.5 shows the way in which the 56-bit key is
used. Initially, the key is passed through a permutation function. Then, for each of
the sixteen rounds, a subkey (Ki) is produced by the combination of a left circular
shift and a permutation. The permutation function is the same for each round, but a
different subkey is produced because of the repeated shifts of the key bits.

DES Decryption

As with any Feistel cipher, decryption uses the same algorithm as encryption, except
that the application of the subkeys is reversed. Additionally, the initial and final
permutations are reversed.

4.3 A DES EXAMPLE

We now work through an example and consider some of its implications. Although
you are not expected to duplicate the example by hand, you will find it informative
to study the hex patterns that occur from one step to the next.

For this example, the plaintext is a hexadecimal palindrome. The plaintext,
key, and resulting ciphertext are as follows:

Plaintext: 02468aceeca86420

Key: 0f1571c947d9e859

Ciphertext: da02ce3a89ecac3b

Results

Table 4.2 shows the progression of the algorithm. The first row shows the 32-bit
values of the left and right halves of data after the initial permutation. The next 16
rows show the results after each round. Also shown is the value of the 48-bit subkey
generated for each round. Note that Li = Ri- 1. The final row shows the left- and
right-hand values after the inverse initial permutation. These two values combined
form the ciphertext.

The Avalanche Effect

A desirable property of any encryption algorithm is that a small change in either
the plaintext or the key should produce a significant change in the ciphertext. In
particular, a change in one bit of the plaintext or one bit of the key should produce

132 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

a change in many bits of the ciphertext. This is referred to as the avalanche effect.
If the change were small, this might provide a way to reduce the size of the plaintext
or key space to be searched.

Using the example from Table 4.2, Table 4.3 shows the result when the fourth
bit of the plaintext is changed, so that the plaintext is 12468aceeca86420. The
second column of the table shows the intermediate 64-bit values at the end of each
round for the two plaintexts. The third column shows the number of bits that differ
between the two intermediate values. The table shows that, after just three rounds,
18 bits differ between the two blocks. On completion, the two ciphertexts differ in
32 bit positions.

Table 4.4 shows a similar test using the original plaintext of with two keys that
differ in only the fourth bit position: the original key, 0f1571c947d9e859, and
the altered key, 1f1571c947d9e859. Again, the results show that about half of
the bits in the ciphertext differ and that the avalanche effect is pronounced after just
a few rounds.

Round Ki Li Ri

IP 5a005a00 3cf03c0f

1 1e030f03080d2930 3cf03c0f bad22845

2 0a31293432242318 bad22845 99e9b723

3 23072318201d0c1d 99e9b723 0bae3b9e

4 05261d3824311a20 0bae3b9e 42415649

5 3325340136002c25 42415649 18b3fa41

6 123a2d0d04262a1c 18b3fa41 9616fe23

7 021f120b1c130611 9616fe23 67117cf2

8 1c10372a2832002b 67117cf2 c11bfc09

9 04292a380c341f03 c11bfc09 887fbc6c

10 2703212607280403 887fbc6c 600f7e8b

11 2826390c31261504 600f7e8b f596506e

12 12071c241a0a0f08 f596506e 738538b8

13 300935393c0d100b 738538b8 c6a62c4e

14 311e09231321182a c6a62c4e 56b0bd75

15 283d3e0227072528 56b0bd75 75e8fd8f

16 2921080b13143025 75e8fd8f 25896490

IP−1 da02ce3a 89ecac3b

Note: DES subkeys are shown as eight 6-bit values in hex format

Table 4.2 DES Example

4.3 / A DES EXAMPLE 133

Table 4.3 Avalanche Effect in DES: Change in Plaintext

Round D

9 c11bfc09887fbc6c
99f911532eed7d94

32

10 887fbc6c600f7e8b
2eed7d94d0f23094

34

11 600f7e8bf596506e
d0f23094455da9c4

37

12 f596506e738538b8
455da9c47f6e3cf3

31

13 738538b8c6a62c4e
7f6e3cf34bc1a8d9

29

14 c6a62c4e56b0bd75
4bc1a8d91e07d409

33

15 56b0bd7575e8fd8f
1e07d4091ce2e6dc

31

16 75e8fd8f25896490
1ce2e6dc365e5f59

32

IP−1 da02ce3a89ecac3b
057cde97d7683f2a

32

Round D

02468aceeca86420
12468aceeca86420

1

1 3cf03c0fbad22845
3cf03c0fbad32845

1

2 bad2284599e9b723
bad3284539a9b7a3

5

3 99e9b7230bae3b9e
39a9b7a3171cb8b3

18

4 0bae3b9e42415649
171cb8b3ccaca55e

34

5 4241564918b3fa41
ccaca55ed16c3653

37

6 18b3fa419616fe23
d16c3653cf402c68

33

7 9616fe2367117cf2
cf402c682b2cefbc

32

8 67117cf2c11bfc09
2b2cefbc99f91153

33

Table 4.4 Avalanche Effect in DES: Change in Key

Round D

02468aceeca86420
02468aceeca86420

0

1 3cf03c0fbad22845
3cf03c0f9ad628c5

3

2 bad2284599e9b723
9ad628c59939136b

11

3 99e9b7230bae3b9e
9939136b768067b7

25

4 0bae3b9e42415649
768067b75a8807c5

29

5 4241564918b3fa41
5a8807c5488dbe94

26

6 18b3fa419616fe23
488dbe94aba7fe53

26

7 9616fe2367117cf2
aba7fe53177d21e4

27

8 67117cf2c11bfc09
177d21e4548f1de4

32

Round D

9 c11bfc09887fbc6c
548f1de471f64dfd

34

10 887fbc6c600f7e8b
71f64dfd4279876c

36

11 600f7e8bf596506e
4279876c399fdc0d

32

12 f596506e738538b8
399fdc0d6d208dbb

28

13 738538b8c6a62c4e
6d208dbbb9bdeeaa

33

14 c6a62c4e56b0bd75
b9bdeeaad2c3a56f

30

15 56b0bd7575e8fd8f
d2c3a56f2765c1fb

27

16 75e8fd8f25896490
2765c1fb01263dc4

30

IP−1 da02ce3a89ecac3b
ee92b50606b62b0b

30

134 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

4.4 THE STRENGTH OF DES

Since its adoption as a federal standard, there have been lingering concerns about
the level of security provided by DES. These concerns, by and large, fall into two
areas: key size and the nature of the algorithm.

The Use of 56-Bit Keys

With a key length of 56 bits, there are 256 possible keys, which is approximately
7.2 * 1016 keys. Thus, on the face of it, a brute-force attack appears impractical.
Assuming that, on average, half the key space has to be searched, a single machine
performing one DES encryption per microsecond would take more than a thousand
years to break the cipher.

However, the assumption of one encryption per microsecond is overly con-
servative. As far back as 1977, Diffie and Hellman postulated that the technology
existed to build a parallel machine with 1 million encryption devices, each of which
could perform one encryption per microsecond [DIFF77]. This would bring the
average search time down to about 10 hours. The authors estimated that the cost
would be about $20 million in 1977 dollars.

With current technology, it is not even necessary to use special, purpose-built
hardware. Rather, the speed of commercial, off-the-shelf processors threaten the
security of DES. A recent paper from Seagate Technology [SEAG08] suggests that
a rate of 1 billion (109) key combinations per second is reasonable for today’s mul-
ticore computers. Recent offerings confirm this. Both Intel and AMD now offer
hardware-based instructions to accelerate the use of AES. Tests run on a contem-
porary multicore Intel machine resulted in an encryption rate of about half a bil-
lion encryptions per second [BASU12]. Another recent analysis suggests that with
contemporary supercomputer technology, a rate of 1013 encryptions per second is
reasonable [AROR12].

With these results in mind, Table 4.5 shows how much time is required for a
brute-force attack for various key sizes. As can be seen, a single PC can break DES in
about a year; if multiple PCs work in parallel, the time is drastically shortened. And
today’s supercomputers should be able to find a key in about an hour. Key sizes of
128 bits or greater are effectively unbreakable using simply a brute-force approach.
Even if we managed to speed up the attacking system by a factor of 1 trillion (1012),
it would still take over 100,000 years to break a code using a 128-bit key.

Fortunately, there are a number of alternatives to DES, the most important of
which are AES and triple DES, discussed in Chapters 6 and 7, respectively.

The Nature of the DES Algorithm

Another concern is the possibility that cryptanalysis is possible by exploiting
the characteristics of the DES algorithm. The focus of concern has been on the
eight substitution tables, or S-boxes, that are used in each iteration (described in
Appendix S). Because the design criteria for these boxes, and indeed for the entire
algorithm, were not made public, there is a suspicion that the boxes were con-
structed in such a way that cryptanalysis is possible for an opponent who knows

4.5 / BLOCK CIPHER DESIGN PRINCIPLES 135

Key Size (bits) Cipher

Number of
Alternative

Keys
Time Required at 109

Decryptions/s

Time Required
at 1013

Decryptions/s

56 DES 256 ≈ 7.2 * 1016 255 ns = 1.125 years 1 hour

128 AES 2128 ≈ 3.4 * 1038 2127 ns = 5.3 * 1021 years 5.3 * 1017 years

168 Triple DES 2168 ≈ 3.7 * 1050 2167 ns = 5.8 * 1033 years 5.8 * 1029 years

192 AES 2192 ≈ 6.3 * 1057 2191 ns = 9.8 * 1040 years 9.8 * 1036 years

256 AES 2256 ≈ 1.2 * 1077 2255 ns = 1.8 * 1060 years 1.8 * 1056 years

26 characters
(permutation)

Monoalphabetic 2! = 4 * 1026 2 * 1026 ns = 6.3 * 109 years 6.3 * 106 years

Table 4.5 Average Time Required for Exhaustive Key Search

the weaknesses in the S-boxes. This assertion is tantalizing, and over the years a
number of regularities and unexpected behaviors of the S-boxes have been discov-
ered. Despite this, no one has so far succeeded in discovering the supposed fatal
weaknesses in the S-boxes.9

Timing Attacks

We discuss timing attacks in more detail in Part Two, as they relate to public-key
algorithms. However, the issue may also be relevant for symmetric ciphers. In
essence, a timing attack is one in which information about the key or the plaintext is
obtained by observing how long it takes a given implementation to perform decryp-
tions on various ciphertexts. A timing attack exploits the fact that an encryption
or decryption algorithm often takes slightly different amounts of time on different
inputs. [HEVI99] reports on an approach that yields the Hamming weight (number
of bits equal to one) of the secret key. This is a long way from knowing the actual
key, but it is an intriguing first step. The authors conclude that DES appears to be
fairly resistant to a successful timing attack but suggest some avenues to explore.
Although this is an interesting line of attack, it so far appears unlikely that this tech-
nique will ever be successful against DES or more powerful symmetric ciphers such
as triple DES and AES.

4.5 BLOCK CIPHER DESIGN PRINCIPLES

Although much progress has been made in designing block ciphers that are cryp-
tographically strong, the basic principles have not changed all that much since the
work of Feistel and the DES design team in the early 1970s. In this section we look
at three critical aspects of block cipher design: the number of rounds, design of the
function F, and key scheduling.

9At least, no one has publicly acknowledged such a discovery.

136 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

Number of Rounds

The cryptographic strength of a Feistel cipher derives from three aspects of the
design: the number of rounds, the function F, and the key schedule algorithm. Let
us look first at the choice of the number of rounds.

The greater the number of rounds, the more difficult it is to perform crypt-
analysis, even for a relatively weak F. In general, the criterion should be that the
number of rounds is chosen so that known cryptanalytic efforts require greater
effort than a simple brute-force key search attack. This criterion was certainly used
in the design of DES. Schneier [SCHN96] observes that for 16-round DES, a dif-
ferential cryptanalysis attack is slightly less efficient than brute force: The differen-
tial cryptanalysis attack requires 255.1 operations,10 whereas brute force requires 255.
If DES had 15 or fewer rounds, differential cryptanalysis would require less effort
than a brute-force key search.

This criterion is attractive, because it makes it easy to judge the strength of
an algorithm and to compare different algorithms. In the absence of a cryptana-
lytic breakthrough, the strength of any algorithm that satisfies the criterion can be
judged solely on key length.

Design of Function F

The heart of a Feistel block cipher is the function F, which provides the element of
confusion in a Feistel cipher. Thus, it must be difficult to “unscramble” the substitu-
tion performed by F. One obvious criterion is that F be nonlinear, as we discussed
previously. The more nonlinear F, the more difficult any type of cryptanalysis will be.
There are several measures of nonlinearity, which are beyond the scope of this
book. In rough terms, the more difficult it is to approximate F by a set of linear
equations, the more nonlinear F is.

Several other criteria should be considered in designing F. We would like the
algorithm to have good avalanche properties. Recall that, in general, this means that
a change in one bit of the input should produce a change in many bits of the output.
A more stringent version of this is the strict avalanche criterion (SAC) [WEBS86],
which states that any output bit j of an S-box (see Appendix S for a discussion of
S-boxes) should change with probability 1/2 when any single input bit i is inverted
for all i, j. Although SAC is expressed in terms of S-boxes, a similar criterion could
be applied to F as a whole. This is important when considering designs that do not
include S-boxes.

Another criterion proposed in [WEBS86] is the bit independence criterion
(BIC), which states that output bits j and k should change independently when any
single input bit i is inverted for all i, j, and k. The SAC and BIC criteria appear to
strengthen the effectiveness of the confusion function.

10Differential cryptanalysis of DES requires 247 chosen plaintext. If all you have to work with is known
plaintext, then you must sort through a large quantity of known plaintext–ciphertext pairs looking for the
useful ones. This brings the level of effort up to 255.1.

Hiva-Network.Com

4.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 137

Key Schedule Algorithm

With any Feistel block cipher, the key is used to generate one subkey for each round.
In general, we would like to select subkeys to maximize the difficulty of deducing
individual subkeys and the difficulty of working back to the main key. No general
principles for this have yet been promulgated.

Adams suggests [ADAM94] that, at minimum, the key schedule should guar-
antee key/ciphertext Strict Avalanche Criterion and Bit Independence Criterion.

4.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

avalanche effect
block cipher
confusion
Data Encryption Standard

(DES)
diffusion

Feistel cipher
irreversible mapping
key
permutation
product cipher
reversible mapping

round
round function
subkey
substitution

Review Questions
4.1 Briefly define a nonsingular transformation.
4.2 What is the difference between a block cipher and a stream cipher?
4.3 Why is it not practical to use an arbitrary reversible substitution cipher of the kind

shown in Table 4.1?
4.4 Briefly define the terms substitution and permutation.
4.5 What is the difference between diffusion and confusion?
4.6 Which parameters and design choices determine the actual algorithm of a Feistel

cipher?
4.7 What are the critical aspects of Feistel cipher design?

Problems

4.1 a. In Section 4.1, under the subsection on the motivation for the Feistel cipher struc-
ture, it was stated that, for a block of n bits, the number of different reversible
mappings for the ideal block cipher is 2n!. Justify.

b. In that same discussion, it was stated that for the ideal block cipher, which allows all
possible reversible mappings, the size of the key is n * 2n bits. But, if there are 2n!
possible mappings, it should take log2 2

n! bits to discriminate among the different
mappings, and so the key length should be log2 2

n!. However, log2 2
n! 6 n * 2n.

Explain the discrepancy.

138 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

4.2 Consider a Feistel cipher composed of sixteen rounds with a block length of 128 bits
and a key length of 128 bits. Suppose that, for a given k, the key scheduling algorithm
determines values for the first eight round keys, k1, k2, c k8, and then sets

k9 = k8, k10 = k7, k11 = k6, c , k16 = k1

Suppose you have a ciphertext c. Explain how, with access to an encryption oracle,
you can decrypt c and determine m using just a single oracle query. This shows that
such a cipher is vulnerable to a chosen plaintext attack. (An encryption oracle can be
thought of as a device that, when given a plaintext, returns the corresponding cipher-
text. The internal details of the device are not known to you and you cannot break
open the device. You can only gain information from the oracle by making queries to
it and observing its responses.)

4.3 Let p be a permutation of the integers 0, 1, 2, c , (2n - 1), such that p(m) gives the
permuted value of m, 0 … m 6 2n. Put another way, p maps the set of n-bit integers
into itself and no two integers map into the same integer. DES is such a permutation
for 64-bit integers. We say that p has a fixed point at m if p(m) = m. That is, if p is
an encryption mapping, then a fixed point corresponds to a message that encrypts to
itself. We are interested in the number of fixed points in a randomly chosen permuta-
tion p. Show the somewhat unexpected result that the number of fixed points for p is
1 on an average, and this number is independent of the size of the permutation.

4.4 Consider a block encryption algorithm that encrypts blocks of length n, and let
N = 2n. Say we have t plaintext–ciphertext pairs Pi, Ci = E(K, Pi), where we assume
that the key K selects one of the N! possible mappings. Imagine that we wish to find K
by exhaustive search. We could generate key K′ and test whether Ci = E(K′, Pi) for
1 … i … t. If K′ encrypts each Pi to its proper Ci, then we have evidence that K = K′.
However, it may be the case that the mappings E(K, # ) and E(K′, # ) exactly agree
on the t plaintext–cipher text pairs Pi, Ci and agree on no other pairs.
a. What is the probability that E(K, # ) and E(K′, # ) are in fact distinct mappings?
b. What is the probability that E(K, # ) and E(K′, # ) agree on another t′ plaintext–

ciphertext pairs where 0 … t′ … N - t?
4.5 For any block cipher, the fact that it is a nonlinear function is crucial to its security. To

see this, suppose that we have a linear block cipher EL that encrypts 256-bit blocks
of plaintext into 256-bit blocks of ciphertext. Let EL(k, m) denote the encryption of a
256-bit message m under a key k (the actual bit length of k is irrelevant). Thus,

EL(k, [m1⊕ m2]) = EL(k, m1)⊕ EL(k, m2) for all [email protected] patterns m1, m2.

Describe how, with 256 chosen ciphertexts, an adversary can decrypt any ciphertext
without knowledge of the secret key k. (A “chosen ciphertext” means that an adver-
sary has the ability to choose a ciphertext and then obtain its decryption. Here, you
have 256 plaintext/ciphertext pairs to work with and you have the ability to choose
the value of the ciphertexts.)

4.6 Suppose the DES F function mapped every 32-bit input R, regardless of the value of
the input K, to;
a. 32-bit string of zero
b. R

Then
1. What function would DES then compute?
2. What would the decryption look like?

Hint: Use the following properties of the XOR operation:

(A⊕ B)⊕ C = A⊕ (B⊕ C)
(A⊕ A) = 0
(A⊕ 0 ) = A

A⊕ 1 = bitwise complement of A

4.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 139

where
A,B,C are n-bit strings of bits
0 is an n-bit string of zeros
1 is an n-bit string of one

4.7 Show that DES decryption is, in fact, the inverse of DES encryption.
4.8 The 32-bit swap after the sixteenth iteration of the DES algorithm is needed to make

the encryption process invertible by simply running the ciphertext back through the
algorithm with the key order reversed. This was demonstrated in the preceding prob-
lem. However, it still may not be entirely clear why the 32-bit swap is needed. To
demonstrate why, solve the following exercises. First, some notation:

A ‘B = the concatenation of the bit strings A and B
Ti(R ‘L) = the transformation defined by the ith iteration of the encryption

algorithm for 1 … I … 16
TDi(R ‘L) = the transformation defined by the ith iteration of the decryption

algorithm for 1 … I … 16
T17(R ‘L) = L ‘R, where this transformation occurs after the sixteenth iteration

of the encryption algorithm

a. Show that the composition TD1(IP(IP
-1(T17(T16(L15 ‘R15))))) is equivalent to the

transformation that interchanges the 32-bit halves, L15 and R15. That is, show that

TD1(IP(IP
-1(T17(T16(L15 ‘R15))))) = R15 ‘L15

b. Now suppose that we did away with the final 32-bit swap in the encryption algo-
rithm. Then we would want the following equality to hold:

TD1(IP(IP
-1(T16(L15 ‘R15)))) = L15 ‘R15

Does it?

Note: The following problems refer to details of DES that are described in Appendix S.

4.9 Consider the substitution defined by row 1 of S-box S1 in Table S.2. Show a block
diagram similar to Figure 4.2 that corresponds to this substitution.

4.10 Compute the bits number 4, 17, 41, and 45 at the output of the first round of the DES
decryption, assuming that the ciphertext block is composed of all ones and the exter-
nal key is composed of all ones.

4.11 This problem provides a numerical example of encryption using a one-round version
of DES. We start with the same bit pattern for the key K and the plaintext, namely:

Hexadecimal notation: 0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary notation: 0000 0001 0010 0011 0100 0101 0110 0111

1000 1001 1010 1011 1100 1101 1110 1111

a. Derive K1, the first-round subkey.
b. Derive L0, R0.
c. Expand R0 to get E[R0], where E[ # ] is the expansion function of Table S.1.
d. Calculate A = E[R0]⊕ K1.
e. Group the 48-bit result of (d) into sets of 6 bits and evaluate the corresponding

S-box substitutions.
f. Concatenate the results of (e) to get a 32-bit result, B.

140 CHAPTER 4 / BLOCK CIPHERS AND THE DATA ENCRYPTION STANDARD

g. Apply the permutation to get P(B).
h. Calculate R1 = P(B)⊕ L0.
i. Write down the ciphertext.

4.12 Analyze the amount of left shifts in the DES key schedule by studying Table S.3 (d).
Is there a pattern? What could be the reason for the choice of these constants?

4.13 When using the DES algorithm for decryption, the 16 keys (K1, K2, c , K16) are
used in reverse order. Therefore, the right-hand side of Figure S.1 is not valid for
decryption. Design a key-generation scheme with the appropriate shift schedule
(analogous to Table S.3d) for the decryption process.

4.14 a. Let X′ be the bitwise complement of X. Prove that if the complement of the
plaintext block is taken and the complement of an encryption key is taken, then
the result of DES encryption with these values is the complement of the original
ciphertext. That is,

If Y = E(K, X)

Then Y′ = E(K′, X′)

Hint: Begin by showing that for any two bit strings of equal length, A and B,
(A⊕ B)′ = A′ ⊕ B.

b. It has been said that a brute-force attack on DES requires searching a key space of
256 keys. Does the result of part (a) change that?

4.15 a. We say that a DES key K is weak if DESK is an involution. Exhibit four weak
keys for DES.

b. We say that a DES key K is semi-weak if it is not weak and if there exists a key K′
such that DESK

- 1 = DESK′. Exhibit four semi-weak keys for DES.

Note: The following problems refer to simplified DES, described in Appendix G.
4.16 Refer to Figure G.3, which explains encryption function for S-DES.

a. How important is the initial permutation IP?
b. How important is the SW function in the middle?

4.17 The equations for the variables q and r for S-DES are defined in the section on
S-DES analysis. Provide the equations for s and t.

4.18 Using S-DES, decrypt the string 01000110 using the key 1010000010 by hand.
Show intermediate results after each function (IP, FK, SW, FK, IP

-1). Then decode
the first 4 bits of the plaintext string to a letter and the second 4 bits to another letter
where we encode A through P in base 2 (i.e., A = 0000, B = 0001, c , P = 1111).
Hint: As a midway check, after the xoring with K2, the string should be 11000001.

Programming Problems

4.19 Create software that can encrypt and decrypt using a general substitution block
cipher.

4.20 Create software that can encrypt and decrypt using S-DES. Test data: use plaintext,
ciphertext, and key of Problem 4.18.

141

5.1 Groups

Groups
Abelian Group
Cyclic Group

5.2 Rings

5.3 Fields

5.4 Finite Fields of the Form GF(p)

Finite Fields of Order p
Finding the Multiplicative Inverse in GF(p)
Summary

5.5 Polynomial Arithmetic

Ordinary Polynomial Arithmetic
Polynomial Arithmetic with Coefficients in Zp
Finding the Greatest Common Divisor
Summary

5.6 Finite Fields of the form GF(2n)

Motivation
Modular Polynomial Arithmetic
Finding the Multiplicative Inverse
Computational Considerations
Using a Generator
Summary

5.7 Key Terms, Review Questions, and Problems

CHAPTER

Finite Fields

142 CHAPTER 5 / FINITE FIELDS

Finite fields have become increasingly important in cryptography. A number of
cryptographic algorithms rely heavily on properties of finite fields, notably the
Advanced Encryption Standard (AES) and elliptic curve cryptography. Other exam-
ples include the message authentication code CMAC and the authenticated encryption
scheme GCM.

This chapter provides the reader with sufficient background on the concepts of
finite fields to be able to understand the design of AES and other cryptographic algo-
rithms that use finite fields. Because students unfamiliar with abstract algebra may find
the concepts behind finite fields somewhat difficult to grasp, we approach the topic in a
way designed to enhance understanding. Our plan of attack is as follows:

1. Fields are a subset of a larger class of algebraic structures called rings, which
are in turn a subset of the larger class of groups. In fact, as shown in Figure 5.1,
both groups and rings can be further differentiated. Groups are defined by
a simple set of properties and are easily understood. Each successive subset
(abelian group, ring, commutative ring, and so on) adds additional properties
and is thus more complex. Sections 5.1 through 5.3 will examine groups, rings,
and fields, successively.

2. Finite fields are a subset of fields, consisting of those fields with a finite num-
ber of elements. These are the class of fields that are found in cryptographic
algorithms. With the concepts of fields in hand, we turn in Section 5.4 to a
specific class of finite fields, namely those with p elements, where p is prime.
Certain asymmetric cryptographic algorithms make use of such fields.

3. A more important class of finite fields, for cryptography, comprises those with
2n elements depicted as fields of the form GF(2n). These are used in a wide
variety of cryptographic algorithms. However, before discussing these fields, we
need to analyze the topic of polynomial arithmetic, which is done in Section 5.5.

4. With all of this preliminary work done, we are able at last, in Section 5.6, to
discuss finite fields of the form GF(2n).

Before proceeding, the reader may wish to review Sections 2.1 through 2.3, which
cover relevant topics in number theory.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Distinguish among groups, rings, and fields.

◆ Define finite fields of the form GF(p).

◆ Explain the differences among ordinary polynomial arithmetic, polynomial
arithmetic with coefficients in Zp, and modular polynomial arithmetic in
GF(2n).

◆ Define finite fields of the form GF(2n).

◆ Explain the two different uses of the mod operator.

5.1 / GROUPS 143

5.1 GROUPS

Groups, rings, and fields are the fundamental elements of a branch of mathematics
known as abstract algebra, or modern algebra. In abstract algebra, we are concerned
with sets on whose elements we can operate algebraically; that is, we can combine
two elements of the set, perhaps in several ways, to obtain a third element of the set.
These operations are subject to specific rules, which define the nature of the set. By
convention, the notation for the two principal classes of operations on set elements is
usually the same as the notation for addition and multiplication on ordinary numbers.
However, it is important to note that, in abstract algebra, we are not limited to ordi-
nary arithmetical operations. All this should become clear as we proceed.

Groups

A group G, sometimes denoted by {G, # }, is a set of elements with a binary opera-
tion denoted by # that associates to each ordered pair (a, b) of elements in G an
element (a # b) in G, such that the following axioms are obeyed:1

(A1) Closure: If a and b belong to G, then a # b is also in G.
(A2) Associative: a # (b # c) = (a # b) # c for all a, b, c in G.

1 The operator # is generic and can refer to addition, multiplication, or some other mathematical operation.

Figure 5.1 Groups, Rings, and Fields

Groups

Abelian groups

Rings

Commutative rings

Integral domains

Fields

Finite
fields

144 CHAPTER 5 / FINITE FIELDS

(A3) Identity element: There is an element e in G such that
a # e = e # a = a for all a in G.

(A4) Inverse element: For each a in G, there is an element a′ in G
such that a # a′ = a′ # a = e.

Let Nn denote a set of n distinct symbols that, for convenience, we represent as
{1, 2, c , n}. A permutation of n distinct symbols is a one-to-one mapping from
Nn to Nn.

2 Define Sn to be the set of all permutations of n distinct symbols. Each
element of Sn is represented by a permutation p of the integers in 1, 2, . . . , n.
It is easy to demonstrate that Sn is a group:

A1: If (p, r∈ Sn), then the composite mapping p # r is formed by per-
muting the elements of r according to the permutation p. For
example, {3, 2, 1} # {1, 3, 2} = {2, 3, 1}. The notation for this map-
ping is explained as follows: The value of the first element of p
indicates which element of r is to be in the first position in p # r; the
value of the second element of p indicates which element of r is
to be in the second position in p # r; and so on. Clearly, p # r∈ Sn.

A2: The composition of mappings is also easily seen to be associative.

A3: The identity mapping is the permutation that does not alter the
order of the n elements. For Sn, the identity element is {1, 2, c , n}.

A4: For any p∈ Sn, the mapping that undoes the permutation defined
by p is the inverse element for p. There will always be such an
inverse. For example {2, 3, 1} # {3, 1, 2} = {1, 2, 3}.

2This is equivalent to the definition of permutation in Chapter 2, which stated that a permutation of a
finite set of elements S is an ordered sequence of all the elements of S, with each element appearing
exactly once.

The set of integers (positive, negative, and 0) under addition is an abelian group.
The set of nonzero real numbers under multiplication is an abelian group. The
set Sn from the preceding example is a group but not an abelian group for n 7 2.

If a group has a finite number of elements, it is referred to as a finite group, and
the order of the group is equal to the number of elements in the group. Otherwise,
the group is an infinite group.

Abelian Group

A group is said to be abelian if it satisfies the following additional condition:

(A5) Commutative: a # b = b # a for all a, b in G.

5.2 / RINGS 145

When the group operation is addition, the identity element is 0; the in-
verse element of a is -a; and subtraction is defined with the following rule:
a - b = a + (-b).

Cyclic Group

We define exponentiation within a group as a repeated application of the group
operator, so that a3 = a # a # a. Furthermore, we define a0 = e as the identity ele-
ment, and a-n = (a′)n, where a′ is the inverse element of a within the group.
A group G is cyclic if every element of G is a power ak (k is an integer) of a fixed
element a∈G. The element a is said to generate the group G or to be a generator
of G. A cyclic group is always abelian and may be finite or infinite.

The additive group of integers is an infinite cyclic group generated by the element
1. In this case, powers are interpreted additively, so that n is the nth power of 1.

5.2 RINGS

A ring R, sometimes denoted by {R, + , * }, is a set of elements with two binary
operations, called addition and multiplication,3 such that for all a, b, c in R the fol-
lowing axioms are obeyed.

(A1–A5) R is an abelian group with respect to addition; that is, R satisfies axioms
A1 through A5. For the case of an additive group, we denote the identity element
as 0 and the inverse of a as -a.
(M1) Closure under multiplication: If a and b belong to R, then ab is also in R.

(M2) Associativity of multiplication: a(bc) = (ab)c for all a, b, c in R.

(M3) Distributive laws: a(b + c) = ab + ac for all a, b, c in R.
(a + b)c = ac + bc for all a, b, c in R.

In essence, a ring is a set of elements in which we can do addition, subtraction
[a - b = a + (-b)], and multiplication without leaving the set.

3Generally, we do not use the multiplication symbol, * , but denote multiplication by the concatenation
of two elements.

With respect to addition and multiplication, the set of all n-square matrices over
the real numbers is a ring.

A ring is said to be commutative if it satisfies the following additional condition:

(M4) Commutativity of multiplication: ab = ba for all a, b in R.

Hiva-Network.Com

146 CHAPTER 5 / FINITE FIELDS

Next, we define an integral domain, which is a commutative ring that obeys
the following axioms.

(M5) Multiplicative identity: There is an element 1 in R such that
a1 = 1a = a for all a in R.

(M6) No zero divisors: If a, b in R and ab = 0, then either a = 0
or b = 0.

Let S be the set of even integers (positive, negative, and 0) under the usual
operations of addition and multiplication. S is a commutative ring. The set of all
n-square matrices defined in the preceding example is not a commutative ring.

The set Zn of integers {0, 1, c , n - 1}, together with the arithmetic oper-
ations modulo n, is a commutative ring (Table 4.3).

Let S be the set of integers (positive, negative, and 0) under the usual operations
of addition and multiplication. S is an integral domain.

Familiar examples of fields are the rational numbers, the real numbers, and the
complex numbers. Note that the set of all integers is not a field, because not every
element of the set has a multiplicative inverse; in fact, only the elements 1 and -1
have multiplicative inverses in the integers.

5.3 FIELDS

A field F, sometimes denoted by {F, + , * }, is a set of elements with two binary
operations, called addition and multiplication, such that for all a, b, c in F the follow-
ing axioms are obeyed.

(A1–M6) F is an integral domain; that is, F satisfies axioms A1 through A5 and
M1 through M6.

(M7) Multiplicative inverse: For each a in F, except 0, there is an element
a-1 in F such that aa-1 = (a-1)a = 1.

In essence, a field is a set of elements in which we can do addition, subtraction,
multiplication, and division without leaving the set. Division is defined with the fol-
lowing rule: a/b = a(b-1).

In gaining insight into fields, the following alternate characterization may be
useful. A field F, denoted by {F, +}, is a set of elements with two binary operations,
called addition and multiplication, such that the following conditions hold:

1. F forms an abelian group with respect to addition.

2. The nonzero elements of F form an abelian group with respect to multiplication.

5.4 / FINITE FIELDS OF THE FORM GF(p) 147

3. The distributive law holds. That is, for all a, b, c in F,

a(b + c) = ab + ac.

(a + b)c = ac + bc

4. Figure 5.2 summarizes the axioms that define groups, rings, and fields.

5.4 FINITE FIELDS OF THE FORM GF(p)

In Section 5.3, we defined a field as a set that obeys all of the axioms of Figure 5.2
and gave some examples of infinite fields. Infinite fields are not of particular inter-
est in the context of cryptography. However, in addition to infinite fields, there are
two types of finite fields, as illustrated in Figure 5.3. Finite fields play a crucial role
in many cryptographic algorithms.

It can be shown that the order of a finite field (number of elements in the
field) must be a power of a prime pn, where n is a positive integer. The finite field
of order pn is generally written GF(pn); GF stands for Galois field, in honor of the
mathematician who first studied finite fields. Two special cases are of interest for
our purposes. For n = 1, we have the finite field GF(p); this finite field has a differ-
ent structure than that for finite fields with n 7 1 and is studied in this section. For
finite fields of the form GF(pn), GF(2n) fields are of particular cryptographic inter-
est, and these are covered in Section 5.6.

Finite Fields of Order p

For a given prime, p, we define the finite field of order p, GF(p), as the set Zp of integers
{0, 1, c , p - 1} together with the arithmetic operations modulo p. Note therefore
that we are using ordinary modular arithmetic to define the operations over these fields.

Figure 5.2 Properties of Groups, Rings, and Fields

(A1) Closure under addition: If a and b belong to S, then a + b is also in S
(A2) Associativity of addition: a + (b + c) = (a + b) + c for all a, b, c in S
(A3) Additive identity: There is an element 0 in R such that

a + 0 = 0 + a = a for all a in S
(A4) Additive inverse: For each a in S there is an element –a in S

such that a + (–a) = (–a) + a = 0

(A5) Commutativity of addition: a + b = b + a for all a, b in S

(M1) Closure under multiplication: If a and b belong to S, then ab is also in S
(M2) Associativity of multiplication: a(bc) = (ab)c for all a, b, c in S
(M3) Distributive laws: a(b + c) = ab + ac for all a, b, c in S

(a + b)c = ac + bc for all a, b, c in S

(M4) Commutativity of multiplication: ab = ba for all a, b in S

(M5) Multiplicative identity: There is an element 1 in S such that
a1 = 1a = a for all a in S

(M6) No zero divisors: If a, b in S and ab = 0, then either
a = 0 or b = 0

(M7) Multiplicative inverse: If a belongs to S and a ≠ 0, there is an
element a –1 in S such that aa –1 = a –1a = 1

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148 CHAPTER 5 / FINITE FIELDS

Recall that we showed in Section 5.2 that the set Zn of integers {0, 1, c , n - 1},
together with the arithmetic operations modulo n, is a commutative ring (Table 2.5).
We further observed that any integer in Zn has a multiplicative inverse if and only if
that integer is relatively prime to n [see discussion of Equation (2.5)].4 If n is prime,
then all of the nonzero integers in Zn are relatively prime to n, and therefore there
exists a multiplicative inverse for all of the nonzero integers in Zn. Thus, for Zp we
can add the following properties to those listed in Table 5.2:

Multiplicative
inverse (w-1)

For each w ∈ Zp, w ≠ 0, there exists a z∈ Zp
such that w * z K 1 (mod p)

Because w is relatively prime to p, if we multiply all the elements of Zp by
w, the resulting residues are all of the elements of Zp permuted. Thus, exactly one
of the residues has the value 1. Therefore, there is some integer in Zp that, when
multiplied by w, yields the residue 1. That integer is the multiplicative inverse of w,
designated w-1. Therefore, Zp is in fact a finite field. Furthermore, Equation (2.5) is
consistent with the existence of a multiplicative inverse and can be rewritten with-
out the condition:

if (a * b) K (a * c)(mod p) then b K c(mod p) (5.1)

Multiplying both sides of Equation (5.1) by the multiplicative inverse of a, we have

((a-1) * a * b) K ((a-1) * a * c)(mod p)
b K c (mod p)

4As stated in the discussion of Equation (2.5), two integers are relatively prime if their only common
positive integer factor is 1.

Figure 5.3 Types of Fields

Fields

Fields with an
infinite number

of elements

Finite fields

GF(p)
Finite fields

with p elements

GF(pn)
Finite fields

with pn elements

The simplest finite field is GF(2). Its arithmetic operations are easily summarized:

+ 0 1
0 0 1

1 1 0

Addition

* 0 1
0 0 0

1 0 1

Multiplication

w -w w-1

0 0 -
1 1 1

Inverses

In this case, addition is equivalent to the exclusive-OR (XOR) operation, and
multiplication is equivalent to the logical AND operation.

5.4 / FINITE FIELDS OF THE FORM GF(p) 149

The right-hand side of Table 5.1 shows arithmetic operations in GF(7). This is a
field of order 7 using modular arithmetic modulo 7. As can be seen, it satisfies all
of the properties required of a field (Figure 5.2). Compare with the left-hand side
of Table 5.1, which reproduces Table 2.2. In the latter case, we see that the set Z8,
using modular arithmetic modulo 8, is not a field. Later in this chapter, we show
how to define addition and multiplication operations on Z8 in such a way as to
form a finite field.

Finding the Multiplicative Inverse in GF(p)

It is easy to find the multiplicative inverse of an element in GF(p) for small values
of p. You simply construct a multiplication table, such as shown in Table 5.1e, and
the desired result can be read directly. However, for large values of p, this approach
is not practical.

If a and b are relatively prime, then b has a multiplicative inverse modulo a.
That is, if gcd(a, b) = 1, then b has a multiplicative inverse modulo a. That is, for
positive integer b 6 a, there exists a b-1 6 a such that bb-1 = 1 mod a. If a is a
prime number and b 6 a, then clearly a and b are relatively prime and have a great-
est common divisor of 1. We now show that we can easily compute b-1 using the
extended Euclidean algorithm.

We repeat here Equation (2.7), which we showed can be solved with the ex-
tended Euclidean algorithm:

ax + by = d = gcd(a, b)

Now, if gcd(a, b) = 1, then we have ax + by = 1. Using the basic equalities of
modular arithmetic, defined in Section 2.3, we can say

[(ax mod a) + (by mod a)] mod a = 1 mod a
0 + (by mod a) = 1

But if by mod a = 1, then y = b-1. Thus, applying the extended Euclidean
algorithm to Equation (2.7) yields the value of the multiplicative inverse of b if
gcd(a, b) = 1.

Consider the example that was shown in Table 2.4. Here we have a = 1759,
which is a prime number, and b = 550. The solution of the equation
1759x + 550y = d yields a value of y = 355. Thus, b-1 = 355. To verify, we cal-
culate 550 * 355 mod 1759 = 195250 mod 1759 = 1.

More generally, the extended Euclidean algorithm can be used to find a
multiplicative inverse in Zn for any n. If we apply the extended Euclidean algorithm
to the equation nx + by = d, and the algorithm yields d = 1, then y = b-1 in Zn.

150 CHAPTER 5 / FINITE FIELDS

+ 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 7

1 1 2 3 4 5 6 7 0

2 2 3 4 5 6 7 0 1

3 3 4 5 6 7 0 1 2

4 4 5 6 7 0 1 2 3

5 5 6 7 0 1 2 3 4

6 6 7 0 1 2 3 4 5

7 7 0 1 2 3 4 5 6

(a) Addition modulo 8

* 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7

2 0 2 4 6 0 2 4 6

3 0 3 6 1 4 7 2 5

4 0 4 0 4 0 4 0 4

5 0 5 2 7 4 1 6 3

6 0 6 4 2 0 6 4 2

7 0 7 6 5 4 3 2 1

(b) Multiplication modulo 8

w 0 1 2 3 4 5 6 7

-w 0 7 6 5 4 3 2 1

w-1 — 1 — 3 — 5 — 7

(c) Additive and multiplicative
inverses modulo 8

+ 0 1 2 3 4 5 6

0 0 1 2 3 4 5 6

1 1 2 3 4 5 6 0

2 2 3 4 5 6 0 1

3 3 4 5 6 0 1 2

4 4 5 6 0 1 2 3

5 5 6 0 1 2 3 4

6 6 0 1 2 3 4 5

(d) Addition modulo 7

* 0 1 2 3 4 5 6

0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6

2 0 2 4 6 1 3 5

3 0 3 6 2 5 1 4

4 0 4 1 5 2 6 3

5 0 5 3 1 6 4 2

6 0 6 5 4 3 2 1

(e) Multiplication modulo 7

w 0 1 2 3 4 5 6

-w 0 6 5 4 3 2 1

w-1 — 1 4 5 2 3 6

(f) Additive and multiplicative
inverses modulo 7

Table 5.1 Arithmetic Modulo 8 and Modulo 7

Summary

In this section, we have shown how to construct a finite field of order p, where p is
prime. Specifically, we defined GF(p) with the following properties.

1. GF(p) consists of p elements.

2. The binary operations + and * are defined over the set. The operations of
addition, subtraction, multiplication, and division can be performed without
leaving the set. Each element of the set other than 0 has a multiplicative in-
verse, and division is performed by multiplication by the multiplicative inverse.

We have shown that the elements of GF(p) are the integers {0, 1, c , p - 1}
and that the arithmetic operations are addition and multiplication mod p.

5.5 / POLYNOMIAL ARITHMETIC 151

5.5 POLYNOMIAL ARITHMETIC

Before continuing our discussion of finite fields, we need to introduce the interest-
ing subject of polynomial arithmetic. We are concerned with polynomials in a single
variable x, and we can distinguish three classes of polynomial arithmetic (Figure 5.4).

■ Ordinary polynomial arithmetic, using the basic rules of algebra.

■ Polynomial arithmetic in which the arithmetic on the coefficients is performed
modulo p; that is, the coefficients are in GF(p).

■ Polynomial arithmetic in which the coefficients are in GF(p), and the poly-
nomials are defined modulo a polynomial m(x) whose highest power is some
integer n.

This section examines the first two classes, and the next section covers the
last class.

Ordinary Polynomial Arithmetic

A polynomial of degree n (integer n Ú 0) is an expression of the form

f(x) = anxn + an - 1xn - 1 + g + a1x + a0 = a
n

i=0
aix

i

where the ai are elements of some designated set of numbers S, called the coefficient
set, and an ≠ 0. We say that such polynomials are defined over the coefficient set S.

A zero-degree polynomial is called a constant polynomial and is simply an
element of the set of coefficients. An nth-degree polynomial is said to be a monic
polynomial if an = 1.

In the context of abstract algebra, we are usually not interested in evaluating a
polynomial for a particular value of x [e.g., f(7)]. To emphasize this point, the vari-
able x is sometimes referred to as the indeterminate.

Polynomial arithmetic includes the operations of addition, subtraction, and
multiplication. These operations are defined in a natural way as though the variable

Figure 5.4 Treatment of Polynomials

Polynomial f(x)

x treated as a variable,
and evaluated for

a particular value of x

x treated as an
indeterminate

Ordinary
polynomial
arithmetic

Arithmetic on
coefficients is

performed
modulo p

Arithmetic on coefficients is
performed modulo p

and polynomials are defined
modulo a polynomial m(x)

152 CHAPTER 5 / FINITE FIELDS

x was an element of S. Division is similarly defined, but requires that S be a field.
Examples of fields include the real numbers, rational numbers, and Zp for p prime.
Note that the set of all integers is not a field and does not support polynomial
division.

Addition and subtraction are performed by adding or subtracting correspond-
ing coefficients. Thus, if

f(x) = a
n

i=0
aix

i; g(x) = a
m

i=0
bix

i; n Ú m

then addition is defined as

f(x) + g(x) = a
m

i=0
(ai + bi)xi + a

n

i=m + 1
aix

i

and multiplication is defined as

f(x) * g(x) = a
n + m

i=0
cix

i

where

ck = a0bk + a1bk - 1 + g + ak - 1b1 + akb0

In the last formula, we treat ai as zero for i 7 n and bi as zero for i 7 m. Note that
the degree of the product is equal to the sum of the degrees of the two polynomials.

As an example, let f(x) = x3 + x2 + 2 and g(x) = x2 - x + 1, where S is the set
of integers. Then

f(x) + g(x) = x3 + 2x2 - x + 3
f(x) - g(x) = x3 + x + 1
f(x) * g(x) = x5 + 3x2 - 2x + 2

Figures 5.5a through 5.5c show the manual calculations. We comment on division
subsequently.

Polynomial Arithmetic with Coefficients in Zp
Let us now consider polynomials in which the coefficients are elements of some
field F; we refer to this as a polynomial over the field F. In this case, it is easy to
show that the set of such polynomials is a ring, referred to as a polynomial ring. That
is, if we consider each distinct polynomial to be an element of the set, then that set
is a ring.5

When polynomial arithmetic is performed on polynomials over a field, then
division is possible. Note that this does not mean that exact division is possible. Let

5In fact, the set of polynomials whose coefficients are elements of a commutative ring forms a polynomial
ring, but that is of no interest in the present context.

5.5 / POLYNOMIAL ARITHMETIC 153

us clarify this distinction. Within a field, given two elements a and b, the quotient
a/b is also an element of the field. However, given a ring R that is not a field, in gen-
eral, division will result in both a quotient and a remainder; this is not exact division.

Figure 5.5 Examples of Polynomial Arithmetic

x3

x3

+ +x2

+2x2
x2 x

2

+–+ ( )

× ( )

– ( )

x–

1

+ 3

(a) Addition

(d) Division(c) Multiplication

x3

x3

+ +x2

+ x2
x2 x

2

x3
x 2

+

+

+x2

x3 – x2

2x2
+ x

– x

x

2

+ 2

2x2– 2x + 2

x4 –– –x3 2x

– 2x

x5 + +x4 2x2

x5 +3x2

+– 1 x2 x +– 1

+ 2

+ 2

x3

x3

+ +x2

x2 x

2

+–

x+

1

+ 1

(b) Subtraction

Consider the division 5/3 within a set S. If S is the set of rational numbers, which
is a field, then the result is simply expressed as 5/3 and is an element of S. Now
suppose that S is the field Z7. In this case, we calculate (using Table 5.1f)

5/3 = (5 * 3-1) mod 7 = (5 * 5) mod 7 = 4
which is an exact solution. Finally, suppose that S is the set of integers, which is a
ring but not a field. Then 5/3 produces a quotient of 1 and a remainder of 2:

5/3 = 1 + 2/3
5 = 1 * 3 + 2

Thus, division is not exact over the set of integers.

Now, if we attempt to perform polynomial division over a coefficient set that
is not a field, we find that division is not always defined.

If the coefficient set is the integers, then (5x2)/(3x) does not have a solution,
because it would require a coefficient with a value of 5/3, which is not in the coef-
ficient set. Suppose that we perform the same polynomial division over Z7. Then
we have (5x2)/(3x) = 4x, which is a valid polynomial over Z7.

However, as we demonstrate presently, even if the coefficient set is a field,
polynomial division is not necessarily exact. In general, division will produce a quo-
tient and a remainder. We can restate the division algorithm of Equation (2.1) for
polynomials over a field as follows. Given polynomials f(x) of degree n and g(x)

154 CHAPTER 5 / FINITE FIELDS

of degree (m), (n Ú m), if we divide f(x) by g(x), we get a quotient q(x) and a
remainder r(x) that obey the relationship

f(x) = q(x)g(x) + r(x) (5.2)

with polynomial degrees:

Degree f(x) = n
Degree g(x) = m
Degree q(x) = n - m
Degree r(x) … m - 1

With the understanding that remainders are allowed, we can say that poly-
nomial division is possible if the coefficient set is a field. One common technique
used for polynomial division is polynomial long division, similar to long division for
integers. Examples of this are shown subsequently.

In an analogy to integer arithmetic, we can write f(x) mod g(x) for the remain-
der r(x) in Equation (5.2). That is, r(x) = f(x) mod g(x). If there is no remainder
[i.e., r(x) = 0], then we can say g(x) divides f(x), written as g(x) � f(x). Equivalently,
we can say that g(x) is a factor of f(x) or g(x) is a divisor of f(x).

For the preceding example [f(x) = x3 + x2 + 2 and g(x) = x2 - x + 1], f(x)/g(x)
produces a quotient of q(x) = x + 2 and a remainder r(x) = x, as shown in
Figure 5.5d. This is easily verified by noting that

q(x)g(x) + r(x) = (x + 2)(x2 - x + 1) + x = (x3 + x2 - x + 2) + x
= x3 + x2 + 2 = f(x)

For our purposes, polynomials over GF(2) are of most interest. Recall from
Section 5.4 that in GF(2), addition is equivalent to the XOR operation, and multi-
plication is equivalent to the logical AND operation. Further, addition and subtrac-
tion are equivalent mod 2:

1 + 1 = 1 - 1 = 0
1 + 0 = 1 - 0 = 1
0 + 1 = 0 - 1 = 1

Figure 5.6 shows an example of polynomial arithmetic over GF(2). For
f(x) = (x7 + x5 + x4 + x3 + x + 1) and g(x) = (x3 + x + 1), the figure shows
f(x) + g(x); f(x) - g(x); f(x) * g(x); and f(x)/g(x). Note that g(x) � f(x).

A polynomial f(x) over a field F is called irreducible if and only if f(x) can-
not be expressed as a product of two polynomials, both over F, and both of degree
lower than that of f(x). By analogy to integers, an irreducible polynomial is also
called a prime polynomial.

The polynomial6 f(x) = x4 + 1 over GF(2) is reducible, because
x4 + 1 = (x + 1)(x3 + x2 + x + 1).

6In the reminder of this chapter, unless otherwise noted, all examples are of polynomials over GF(2).

Hiva-Network.Com

5.5 / POLYNOMIAL ARITHMETIC 155

Consider the polynomial f(x) = x3 + x + 1. It is clear by inspection that x is not
a factor of f(x). We easily show that x + 1 is not a factor of f(x):

x2 + x
x + 1�x3 + x + 1

x3 + x2

x2 + x
x2 + x

1
Thus, f(x) has no factors of degree 1. But it is clear by inspection that if f(x) is
reducible, it must have one factor of degree 2 and one factor of degree 1. There-
fore, f(x) is irreducible.

Figure 5.6 Examples of Polynomial Arithmetic over GF(2)

(a) Addition

(c) Multiplication

(d) Division

x4x5 ++x7
xx3

x3x4 ++x5 ++x7 +x 1

+++ ( )1

x3x4 ++x5 ++x7 +x 1

x4x5 ++x7
x3 x

x3 ++ +x 1

+ 1

x5x6 ++x8 x4 ++ +x2

+ x2

x

x7x8 ++x10 x6 ++ +x4

x10 + x4
x3

++× ( )1

x3x4 ++x5 ++x7

x4x5 ++x7

+x

x3 x

1

++– ( )1

(b) Subtraction

x3x4 ++x5 ++

++

x7

x4x5x7
+x 1

x3 + +x 1

x3 + +x 1

x4 1+

x3 x ++ 1

156 CHAPTER 5 / FINITE FIELDS

Finding the Greatest Common Divisor

We can extend the analogy between polynomial arithmetic over a field and integer
arithmetic by defining the greatest common divisor as follows. The polynomial c(x)
is said to be the greatest common divisor of a(x) and b(x) if the following are true.

1. c(x) divides both a(x) and b(x).

2. Any divisor of a(x) and b(x) is a divisor of c(x).

An equivalent definition is the following: gcd[a(x), b(x)] is the polynomial of
maximum degree that divides both a(x) and b(x).

We can adapt the Euclidean algorithm to compute the greatest common divisor
of two polynomials. Recall Equation (2.6), from Chapter 2, which is the basis of the
Euclidean algorithm: gcd(a, b) = gcd(b, a mod b). This equality can be rewritten as the
following equation:

gcd[a(x), b(x)] = gcd[b(x), a(x) mod b(x)] (5.3)

Equation (5.3) can be used repetitively to determine the greatest common divisor.
Compare the following scheme to the definition of the Euclidean algorithm for integers.

Euclidean Algorithm for Polynomials

Calculate Which satisfies

r1(x) = a(x) mod b(x) a(x) = q1(x)b(x) + r1(x)
r2(x) = b(x) mod r1(x) b(x) = q2(x)r1(x) + r2(x)
r3(x) = r1(x) mod r2(x) r1(x) = q3(x)r2(x) + r3(x)

rn(x) = rn - 2(x) mod rn - 1(x) rn - 2(x) = qn(x)rn - 1(x) + rn(x)

rn + 1(x) = rn - 1(x) mod rn(x) = 0
rn - 1(x) = qn + 1(x)rn(x) + 0

d(x) = gcd(a(x), b(x)) = rn(x)

At each iteration, we have d(x) = gcd(ri+ 1(x), ri(x)) until finally
d(x) = gcd(rn(x), 0) = rn(x). Thus, we can find the greatest common divisor of two
integers by repetitive application of the division algorithm. This is the Euclidean
algorithm for polynomials. The algorithm assumes that the degree of a(x) is greater
than the degree of b(x).

Find gcd[a(x), b(x)] for a(x) = x6 + x5 + x4 + x3 + x2 + x + 1 and b(x) =
x4 + x2 + x + 1. First, we divide a(x) by b(x):

x2 + x
x4 + x2 + x + 1�x6 + x5 + x4 + x3 + x2 + x + 1

x6 + x4 + x3 + x2

x5 + x + 1
x5 + x3 + x2 + x

x3 + x2 + 1

5.6 / FINITE FIELDS OF THE FORM GF(2n) 157

Summary

We began this section with a discussion of arithmetic with ordinary polynomials. In
ordinary polynomial arithmetic, the variable is not evaluated; that is, we do not plug
a value in for the variable of the polynomials. Instead, arithmetic operations are
performed on polynomials (addition, subtraction, multiplication, division) using the
ordinary rules of algebra. Polynomial division is not allowed unless the coefficients
are elements of a field.

Next, we discussed polynomial arithmetic in which the coefficients are ele-
ments of GF(p). In this case, polynomial addition, subtraction, multiplication, and
division are allowed. However, division is not exact; that is, in general division re-
sults in a quotient and a remainder.

Finally, we showed that the Euclidean algorithm can be extended to find the
greatest common divisor of two polynomials whose coefficients are elements of a
field.

All of the material in this section provides a foundation for the following sec-
tion, in which polynomials are used to define finite fields of order pn.

5.6 FINITE FIELDS OF THE FORM GF(2n)

Earlier in this chapter, we mentioned that the order of a finite field must be of the
form pn, where p is a prime and n is a positive integer. In Section 5.4, we looked at
the special case of finite fields with order p. We found that, using modular arith-
metic in Zp, all of the axioms for a field (Figure 5.2) are satisfied. For polynomials
over pn, with n 7 1, operations modulo pn do not produce a field. In this section,
we show what structure satisfies the axioms for a field in a set with pn elements and
concentrate on GF(2n).

Motivation

Virtually all encryption algorithms, both symmetric and asymmetric, involve arith-
metic operations on integers. If one of the operations that is used in the algorithm is
division, then we need to work in arithmetic defined over a field. For convenience

This yields r1(x) = x3 + x2 + 1 and q1 (x) = x2 + x.
Then, we divide b(x) by r1(x).

x + 1
x3 + x2 + 1�x4 + x2 + x + 1

x4 + x3 + x
x3 + x2 + 1
x3 + x2 + 1

This yields r2(x) = 0 and q2(x) = x + 1.
Therefore, gcd[a(x), b(x)] = r1(x) = x3 + x2 + 1.

158 CHAPTER 5 / FINITE FIELDS

and for implementation efficiency, we would also like to work with integers that fit
exactly into a given number of bits with no wasted bit patterns. That is, we wish to
work with integers in the range 0 through 2n - 1, which fit into an n-bit word.

Suppose we wish to define a conventional encryption algorithm that operates on
data 8 bits at a time, and we wish to perform division. With 8 bits, we can repre-
sent integers in the range 0 through 255. However, 256 is not a prime number, so
that if arithmetic is performed in Z256 (arithmetic modulo 256), this set of inte-
gers will not be a field. The closest prime number less than 256 is 251. Thus, the
set Z251, using arithmetic modulo 251, is a field. However, in this case the 8-bit
patterns representing the integers 251 through 255 would not be used, resulting
in inefficient use of storage.

As the preceding example points out, if all arithmetic operations are to be
used and we wish to represent a full range of integers in n bits, then arithmetic
modulo 2n will not work. Equivalently, the set of integers modulo 2n for n 7 1, is
not a field. Furthermore, even if the encryption algorithm uses only addition and
multiplication, but not division, the use of the set Z2n is questionable, as the follow-
ing example illustrates.

Suppose we wish to use 3-bit blocks in our encryption algorithm and use only the
operations of addition and multiplication. Then arithmetic modulo 8 is well defined,
as shown in Table 5.1. However, note that in the multiplication table, the nonzero
integers do not appear an equal number of times. For example, there are only four
occurrences of 3, but twelve occurrences of 4. On the other hand, as was mentioned,
there are finite fields of the form GF(2n), so there is in particular a finite field of
order 23 = 8. Arithmetic for this field is shown in Table 5.2. In this case, the number
of occurrences of the nonzero integers is uniform for multiplication. To summarize,

Integer 1 2 3 4 5 6 7
Occurrences in Z8 4 8 4 12 4 8 4
Occurrences in GF(23) 7 7 7 7 7 7 7

For the moment, let us set aside the question of how the matrices of Table 5.2
were constructed and instead make some observations.

1. The addition and multiplication tables are symmetric about the main diago-
nal, in conformance to the commutative property of addition and multiplica-
tion. This property is also exhibited in Table 5.1, which uses mod 8 arithmetic.

2. All the nonzero elements defined by Table 5.2 have a multiplicative inverse,
unlike the case with Table 5.1.

3. The scheme defined by Table 5.2 satisfies all the requirements for a finite
field. Thus, we can refer to this scheme as GF(23).

4. For convenience, we show the 3-bit assignment used for each of the elements
of GF(23).

5.6 / FINITE FIELDS OF THE FORM GF(2n) 159

Intuitively, it would seem that an algorithm that maps the integers unevenly
onto themselves might be cryptographically weaker than one that provides a uni-
form mapping. That is, a cryptanalytic technique might be able to exploit the fact
that some integers occur more frequently and some less frequently in the ciphertext.
Thus, the finite fields of the form GF(2n) are attractive for cryptographic algorithms.

To summarize, we are looking for a set consisting of 2n elements, together
with a definition of addition and multiplication over the set that define a field. We
can assign a unique integer in the range 0 through 2n - 1 to each element of the
set. Keep in mind that we will not use modular arithmetic, as we have seen that this
does not result in a field. Instead, we will show how polynomial arithmetic provides
a means for constructing the desired field.

Modular Polynomial Arithmetic

Consider the set S of all polynomials of degree n - 1 or less over the field Zp. Thus,
each polynomial has the form

f(x) = an - 1xn - 1 + an - 2xn - 2 + g + a1x + a0 = a
n - 1

i=0
aix

i

000 001 010 011 100 101 110 111

+ 0 1 2 3 4 5 6 7

000 0 0 1 2 3 4 5 6 7

001 1 1 0 3 2 5 4 7 6

010 2 2 3 0 1 6 7 4 5

011 3 3 2 1 0 7 6 5 4

100 4 4 5 6 7 0 1 2 3

101 5 5 4 7 6 1 0 3 2

110 6 6 7 4 5 2 3 0 1

111 7 7 6 5 4 3 2 1 0

(a) Addition

000 001 010 011 100 101 110 111

* 0 1 2 3 4 5 6 7

000 0 0 0 0 0 0 0 0 0

001 1 0 1 2 3 4 5 6 7

010 2 0 2 4 6 3 1 7 5

011 3 0 3 6 5 7 4 1 2

100 4 0 4 3 7 6 2 5 1

101 5 0 5 1 4 2 7 3 6

110 6 0 6 7 1 5 3 2 4

111 7 0 7 5 2 1 6 4 3

(b) Multiplication

w -w w-1

0 0 -

1 1 1

2 2 5

3 3 6

4 4 7

5 5 2

6 6 3

7 7 4

(c) Additive and multiplicative
inverses

Table 5.2 Arithmetic in GF(23)

160 CHAPTER 5 / FINITE FIELDS

where each ai takes on a value in the set {0, 1, c , p - 1}. There are a total of pn
different polynomials in S.

For p = 3 and n = 2, the 32 = 9 polynomials in the set are
0, 1, 2, x, x + 1, x + 2, 2x, 2x + 1, 2x + 2

For p = 2 and n = 3, the 23 = 8 polynomials in the set are
0, 1, x, x + 1, x2, x2 + 1, x2 + x, x2 + x + 1

With the appropriate definition of arithmetic operations, each such set S is a
finite field. The definition consists of the following elements.

1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic
rules of algebra, with the following two refinements.

2. Arithmetic on the coefficients is performed modulo p. That is, we use the rules
of arithmetic for the finite field Zp.

3. If multiplication results in a polynomial of degree greater than n - 1, then the
polynomial is reduced modulo some irreducible polynomial m(x) of degree n.
That is, we divide by m(x) and keep the remainder. For a polynomial f(x), the
remainder is expressed as r(x) = f(x) mod m(x).

The Advanced Encryption Standard (AES) uses arithmetic in the finite field
GF(28), with the irreducible polynomial m(x) = x8 + x4 + x3 + x + 1. Consider
the two polynomials f(x) = x6 + x4 + x2 + x + 1 and g(x) = x7 + x + 1. Then

f(x) + g(x) = x6 + x4 + x2 + x + 1 + x7 + x + 1
= x7 + x6 + x4 + x2

f(x) * g(x) = x13 + x11 + x9 + x8 + x7

+ x7 + x5 + x3 + x2 + x
+ x6 + x4 + x2 + x + 1

= x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1

x5 + x3

x8 + x4 + x3 + x + 1 >x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1
x13 + x9 + x8 + x6 + x5

x11 + x4 + x3

x11 + x7 + x6 + x4 + x3

x7 + x6 + 1

Therefore, f(x) * g(x) mod m(x) = x7 + x6 + 1.

5.6 / FINITE FIELDS OF THE FORM GF(2n) 161

As with ordinary modular arithmetic, we have the notion of a set of residues
in modular polynomial arithmetic. The set of residues modulo m(x), an nth-degree
polynomial, consists of pn elements. Each of these elements is represented by one of
the pn polynomials of degree m 6 n.

The residue class [x + 1], (mod m(x)), consists of all polynomials a(x) such that
a(x) K (x + 1)(mod m(x)). Equivalently, the residue class [x + 1] consists of all
polynomials a(x) that satisfy the equality a(x) mod m(x) = x + 1.

It can be shown that the set of all polynomials modulo an irreducible nth-
degree polynomial m(x) satisfies the axioms in Figure 5.2, and thus forms a finite
field. Furthermore, all finite fields of a given order are isomorphic; that is, any two
finite-field structures of a given order have the same structure, but the representa-
tion or labels of the elements may be different.

To construct the finite field GF(23), we need to choose an irreducible poly-
nomial of degree 3. There are only two such polynomials: (x3 + x2 + 1) and
(x3 + x + 1). Using the latter, Table 5.3 shows the addition and multiplication
tables for GF(23). Note that this set of tables has the identical structure to those
of Table 5.2. Thus, we have succeeded in finding a way to define a field of order 23.

We can now read additions and multiplications from the table easily. For exam-
ple, consider binary 100 + 010 = 110. This is equivalent to x2 + x. Also consider
100 * 010 = 011, which is equivalent to x2 * x = x3 and reduces to x + 1. That
is, x3 mod (x3 + x + 1) = x + 1, which is equivalent to 011.

Finding the Multiplicative Inverse

Just as the Euclidean algorithm can be adapted to find the greatest common divisor
of two polynomials, the extended Euclidean algorithm can be adapted to find the
multiplicative inverse of a polynomial. Specifically, the algorithm will find the mul-
tiplicative inverse of b(x) modulo a(x) if the degree of b(x) is less than the degree of
a(x) and gcd[a(x), b(x)] = 1. If a(x) is an irreducible polynomial, then it has no fac-
tor other than itself or 1, so that gcd[a(x), b(x)] = 1. The algorithm can be charac-
terized in the same way as we did for the extended Euclidean algorithm for integers.
Given polynomials a(x) and b(x) with the degree of a(x) greater than the degree
of b(x), we wish to solve the following equation for the values v(x), w(x), and d(x),
where d(x) = gcd[a(x), b(x)]:

a(x)v(x) + b(x)w(x) = d(x)

If d(x) = 1, then w(x) is the multiplicative inverse of b(x) modulo a(x). The calcula-
tions are as follows.

162 CHAPTER 5 / FINITE FIELDS

00
0

00
1

01
0

01
1

10
0

10
1

11
0

11
1

+
0

1
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x2

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ly

no
m

ia
l A

ri
th

m
et

ic
M

od
ul

o
(x

3
+

x
+

1)

5.6 / FINITE FIELDS OF THE FORM GF(2n) 163

Extended Euclidean Algorithm for Polynomials

Calculate Which satisfies Calculate Which satisfies

r-1(x) = a(x) v-1(x) = 1; w-1(x) = 0 a(x) = a(x)v-1(x) +
bw-1(x)

r0(x) = b(x) v0(x) = 0; w0(x) = 1 b(x) = a(x)v0(x) +
b(x)w0(x)

r1(x) = a(x) mod b(x)
q1(x) = quotient of
a(x)/b(x)

a(x) = q1(x)b(x) +
r1(x)

v1(x) = v-1(x) -
q1(x)v0(x) = 1
w1(x) = w-1(x) -
q1(x)w0(x) = -q1(x)

r1(x) = a(x)v1(x) +
b(x)w1(x)

r2(x) = b(x) mod r1(x)
q2(x) = quotient of
b(x)/r1(x)

b(x) = q2(x)r1(x) +
r2(x)

v2(x) = v0(x) -
q2(x)v1(x)
w2(x) = w0(x) -
q2(x)w1(x)

r2(x) = a(x)v2(x) +
b(x)w2(x)

r3(x) = r1(x) mod r2(x)
q3(x) = quotient of
r1(x)/r2(x)

r1(x) = q3(x)r2(x) +
r3(x)

v3(x) = v1(x) -
q3(x)v2(x)
w3(x) = w1(x) -
q3(x)w2(x)

r3(x) = a(x)v3(x) +
b(x)w3(x)

f

rn(x) = rn - 2(x)
mod rn - 1(x)
qn(x) = quotient of
rn - 2(x)/rn - 2(x)

rn - 2(x) = qn(x)rn - 1(x)
+ rn(x)

vn(x) = vn - 2(x) -
qn(x)vn - 1(x)
wn(x) = wn - 2(x) -
qn(x)wn - 1(x)

rn(x) = a(x)vn(x) +
b(x)wn(x)

rn + 1(x) = rn - 1(x)
mod rn(x) = 0
qn + 1(x) = quotient of
rn - 1(x)/rn(x)

rn - 1(x) = qn + 1(x)rn(x)
+ 0

d(x) = gcd(a(x),
b(x)) = rn(x)
v(x) = vn(x); w(x) =
wn(x)

Table 5.4 shows the calculation of the multiplicative inverse of (x7 + x + 1)
mod (x8 + x4 + x3 + x + 1). The result is that (x7 + x + 1)-1 = (x7). That is,
(x7 + x + 1)(x7) K 1(mod (x8 + x4 + x3 + x + 1)).

Computational Considerations

A polynomial f(x) in GF(2n)

f(x) = an - 1xn - 1 + an - 2xn - 2 + g + a1x + a0 = a
n - 1

i=0
aix

i

can be uniquely represented by the sequence of its n binary coefficients
(an - 1, an - 2, c , a0). Thus, every polynomial in GF(2n) can be represented by an
n-bit number.

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164 CHAPTER 5 / FINITE FIELDS

ADDITION We have seen that addition of polynomials is performed by adding cor-
responding coefficients, and, in the case of polynomials over Z2, addition is just the
XOR operation. So, addition of two polynomials in GF(2n) corresponds to a bitwise
XOR operation.

Initialization a(x) = x8 + x4 + x3 + x + 1; v-1(x) = 1; w-1(x) = 0
b(x) = x7 + x + 1; v0(x) = 0; w0(x) = 1

Iteration 1 q1(x) = x; r1(x) = x4 + x3 + x2 + 1
v1(x) = 1; w1(x) = x

Iteration 2 q2(x) = x3 + x2 + 1; r2(x) = x
v2(x) = x3 + x2 + 1; w2(x) = x4 + x3 + x + 1

Iteration 3 q3(x) = x3 + x2 + x; r3(x) = 1
v3(x) = x6 + x2 + x + 1; w3(x) = x7

Iteration 4 q4(x) = x; r4(x) = 0
v4(x) = x7 + x + 1; w4(x) = x8 + x4 + x3 + x + 1

Result d(x) = r3(x) = gcd(a(x), b(x)) = 1
w(x) = w3(x) = (x7 + x + 1)-1 mod (x8 + x4 + x3 + x + 1) = x7

Table 5.4 Extended Euclid [(x8 + x4 + x3 + x + 1), (x7 + x + 1)]

Tables 5.2 and 5.3 show the addition and multiplication tables for GF(23) modulo
m(x) = (x3 + x + 1). Table 5.2 uses the binary representation, and Table 5.3
uses the polynomial representation.

Consider the two polynomials in GF(28) from our earlier example:

f(x) = x6 + x4 + x2 + x + 1 and g(x) = x7 + x + 1.

(x6 + x4 + x2 + x + 1) + (x7 + x + 1) = x7 + x6 + x4 + x2 (polynomial notation)
(01010111)⊕ (10000011) = (11010100) (binary notation)
{57}⊕ {83} = {D4} (hexadecimal notation)7

7A basic refresher on number systems (decimal, binary, hexadecimal) can be found at the Computer
Science Student Resource Site at WilliamStallings.com/StudentSupport.html. Here each of two groups
of 4 bits in a byte is denoted by a single hexadecimal character, and the two characters are enclosed in
brackets.

MULTIPLICATION There is no simple XOR operation that will accomplish multi-
plication in GF(2n). However, a reasonably straightforward, easily implemented
technique is available. We will discuss the technique with reference to GF(28) using
m(x) = x8 + x4 + x3 + x + 1, which is the finite field used in AES. The technique
readily generalizes to GF(2n).

The technique is based on the observation that

x8 mod m(x) = [m(x) - x8] = (x4 + x3 + x + 1) (5.4)

5.6 / FINITE FIELDS OF THE FORM GF(2n) 165

A moment’s thought should convince you that Equation (5.4) is true; if you
are not sure, divide it out. In general, in GF(2n) with an nth-degree polynomial p(x),
we have xn mod p(x) = [p(x) - xn].

Now, consider a polynomial in GF(28), which has the form
f(x) = b7x7 + b6x6 + b5x5 + b4x4 + b3x3 + b2x2 + b1x + b0. If we multiply by x,
we have

x * f(x) = (b7x8 + b6x7 + b5x6 + b4x5 + b3x4

+ b2x3 + b1x2 + b0x) mod m(x) (5.5)

If b7 = 0, then the result is a polynomial of degree less than 8, which is already
in reduced form, and no further computation is necessary. If b7 = 1, then reduction
modulo m(x) is achieved using Equation (5.4):

x * f(x) = (b6x7 + b5x6 + b4x5 + b3x4 + b2x3 + b1x2 + b0x)
+ (x4 + x3 + x + 1)

It follows that multiplication by x (i.e., 00000010) can be implemented as a 1-bit
left shift followed by a conditional bitwise XOR with (00011011), which represents
(x4 + x3 + x + 1). To summarize,

x * f(x) = b (b6b5b4b3b2b1b00) if b7 = 0
(b6b5b4b3b2b1b00)⊕ (00011011) if b7 = 1

(5.6)

Multiplication by a higher power of x can be achieved by repeated application
of Equation (5.6). By adding intermediate results, multiplication by any constant in
GF(28) can be achieved.

In an earlier example, we showed that for f(x) = x6 + x4 + x2 + x + 1, g(x) = x7 +
x + 1, and m(x) = x8 + x4 + x3 + x + 1, we have f(x) * g(x) mod m(x) = x7 + x6 + 1.
Redoing this in binary arithmetic, we need to compute (01010111) * (10000011). First,
we determine the results of multiplication by powers of x:

(01010111) * (00000010) = (10101110)
(01010111) * (00000100) = (01011100)⊕ (00011011) = (01000111)
(01010111) * (00001000) = (10001110)
(01010111) * (00010000) = (00011100)⊕ (00011011) = (00000111)
(01010111) * (00100000) = (00001110)
(01010111) * (01000000) = (00011100)
(01010111) * (10000000) = (00111000)

So,

(01010111) * (10000011) = (01010111) * [(00000001)⊕ (00000010)⊕ (10000000)]

= (01010111)⊕ (10101110)⊕ (00111000) = (11000001)

which is equivalent to x7 + x6 + 1.

166 CHAPTER 5 / FINITE FIELDS

Using a Generator

An equivalent technique for defining a finite field of the form GF(2n), using the
same irreducible polynomial, is sometimes more convenient. To begin, we need two
definitions: A generator g of a finite field F of order q (contains q elements) is an
element whose first q - 1 powers generate all the nonzero elements of F. That is,
the elements of F consist of 0, g0, g1, c , gq - 2. Consider a field F defined by a
polynomial f(x). An element b contained in F is called a root of the polynomial if
f(b) = 0. Finally, it can be shown that a root g of an irreducible polynomial is a gen-
erator of the finite field defined on that polynomial.

Power
Representation

Polynomial
Representation

Binary
Representation

Decimal (Hex)
Representation

0 0 000 0

g0(= g7) 1 001 1

g1 g 010 2

g2 g2 100 4

g3 g + 1 011 3

g4 g2 + g 110 6

g5 g2 + g + 1 111 7

g6 g2 + 1 101 5

Table 5.5 Generator for GF(23) using x3 + x + 1

Let us consider the finite field GF(23), defined over the irreducible poly-
nomial x3 + x + 1, discussed previously. Thus, the generator g must satisfy
f(g) = g3 + g + 1 = 0. Keep in mind, as discussed previously, that we need not
find a numerical solution to this equality. Rather, we deal with polynomial arith-
metic in which arithmetic on the coefficients is performed modulo 2. Therefore,
the solution to the preceding equality is g3 = -g - 1 = g + 1. We now show
that g in fact generates all of the polynomials of degree less than 3. We have the
following.

g4 = g(g3) = g(g + 1) = g2 + g
g5 = g(g4) = g(g2 + g) = g3 + g2 = g2 + g + 1
g6 = g(g5) = g(g2 + g + 1) = g3 + g2 + g = g2 + g + g + 1 = g2 + 1
g7 = g(g6) = g(g2 + 1) = g3 + g = g + g + 1 = 1 = g0

We see that the powers of g generate all the nonzero polynomials in GF(23).
Also, it should be clear that gk = gk mod7 for any integer k. Table 5.5 shows the
power representation, as well as the polynomial and binary representations.

5.6 / FINITE FIELDS OF THE FORM GF(2n) 167

In general, for GF(2n) with irreducible polynomial f(x), determine
gn = f(g) - gn. Then calculate all of the powers of g from gn + 1 through g2

n - 2.
The elements of the field correspond to the powers of g from g0 through g2

n - 2
plus the value 0. For multiplication of two elements in the field, use the equality
gk = gk mod(2

n - 1) for any integer k.

Summary

In this section, we have shown how to construct a finite field of order 2n. Specifically,
we defined GF(2n) with the following properties.

1. GF(2n) consists of 2n elements.

2. The binary operations + and * are defined over the set. The operations
of addition, subtraction, multiplication, and division can be performed with-
out leaving the set. Each element of the set other than 0 has a multiplicative
inverse.

We have shown that the elements of GF(2n) can be defined as the set of all
polynomials of degree n - 1 or less with binary coefficients. Each such polynomial
can be represented by a unique n-bit value. Arithmetic is defined as polynomial
arithmetic modulo some irreducible polynomial of degree n. We have also seen that
an equivalent definition of a finite field GF(2n) makes use of a generator and that
arithmetic is defined using powers of the generator.

This power representation makes multiplication easy. To multiply in the
power notation, add exponents modulo 7. For example, g4 * g6 = g(10 mod 7) =
g3 = g + 1. The same result is achieved using polynomial arithmetic: We have
g4 = g2 + g and g6 = g2 + 1. Then, (g2 + g) * (g2 + 1) = g4 + g3 + g2 + g.
Next, we need to determine (g4 + g3 + g2 + 1) mod (g3 + g + 1) by division:

g + 1
g3 + g + 1�g4 + g3 + g2 + g

g4 + g2 + g
g3

g3 + g + 1
g + 1

We get a result of g + 1, which agrees with the result obtained using the power
representation.

Table 5.6 shows the addition and multiplication tables for GF(23) using
the power representation. Note that this yields the identical results to the
polynomial representation (Table 5.3) with some of the rows and columns
i nterchanged.

168 CHAPTER 5 / FINITE FIELDS

00
0

00
1

01
0

10
0

01
1

11
0

11
1

10
1

+
0

1
G

g2
g3

g4
g5

g6

00
0

0
0

1
G

g2
g

+
1

g2
+

g
g2

+
g

+
1

g2
+

1

00
1

1
1

0
g

+
1

g2
+

1
g

g2
+

g
+

1
g2

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g2

01
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g

g
+

1
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+

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1

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+

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g2
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g2
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11
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5.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 169

5.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

abelian group
associative
coefficient set
commutative
commutative ring
cyclic group
divisor
Euclidean algorithm
field
finite field
finite group
generator

greatest common divisor
group
identity element
infinite field
infinite group
integral domain
inverse element
irreducible polynomial
modular arithmetic
modular polynomial

arithmetic
monic polynomial

order
polynomial
polynomial arithmetic
polynomial ring
prime number
prime polynomial
relatively prime
residue
ring

Review Questions

5.1 Briefly define a group.
5.2 Briefly define a ring.
5.3 Briefly define a field.
5.4 List three classes of polynomial arithmetic.

Problems

5.1 For the group Sn of all permutations of n distinct symbols,
a. what is the number of elements in Sn?
b. show that Sn is not abelian for n 7 2.

5.2 Does the set of residue classes (mod3) form a group
a. with respect to modular addition?
b. with respect to modular multiplication?

5.3 Let S = {0, a, b, c}. The addition and multiplication on the set S is defined in the
following tables:

+ 0 a B C
0 0 a B C

A a 0 c B

B b c 0 A

C c b a 0

* 0 a b c
0 0 0 0 0

a 0 a b c

b 0 a b c

c 0 0 0 0

Is S a noncommutative ring? Justify your answer.
5.4 Develop a set of tables similar to Table 5.1 for GF(5).
5.5 Demonstrate that the set of polynomials whose coefficients form a field is a ring.
5.6 Demonstrate whether each of these statements is true or false for polynomials over a

field.

170 CHAPTER 5 / FINITE FIELDS

a. The product of monic polynomials is monic.
b. The product of polynomials of degrees m and n has degree m + n.
c. The sum of polynomials of degrees m and n has degree max [m, n].

5.7 For polynomial arithmetic with coefficients in Z1 1, perform the following calculations.
a. (x 2 + 2 x + 9 )(x 3 + 1 1 x 2 + x + 7 )
b. (8 x 2 + 3 x + 2 )(5 x 2 + 6 )

5.8 Determine which of the following polynomials are reducible over GF(2).
a. x 2 + 1
b. x 2 + x + 1
c. x 4 + x + 1

5.9 Determine the gcd of the following pairs of polynomials.
a. (x3 + 1) and (x2 + x + 1) over GF(2)
b. (x3 + x + 1) and (x2 + 1) over GF(3)
c. (x3 - 2x + 1) and (x2 - x - 2) over GF(5)
d. (x4 + 8x3 + 7x + 8) and (2x3 + 9x2 + 10x + 1) over GF(11)

5.10 Develop a set of tables similar to Table 5.3 for GF(3) with m(x) = x2 + x + 1.
5.11 Determine the multiplicative inverse of x 2 + 1 in GF(23) with m(x) = x 3 + x - 1 .
5.12 Develop a table similar to Table 5.5 for GF(25) with m(x) = x 5 + x 4 + x 3 + x + 1 .

Programming Problems

5.13 Write a simple four-function calculator in GF(24). You may use table lookups for the
multiplicative inverses.

5.14 Write a simple four-function calculator in GF(28). You should compute the multiplica-
tive inverses on the fly.

171

6.1 Finite Field Arithmetic

6.2 AES Structure

General Structure
Detailed Structure

6.3 AES Transformation Functions

Substitute Bytes Transformation
ShiftRows Transformation
MixColumns Transformation
AddRoundKey Transformation

6.4 AES Key Expansion

Key Expansion Algorithm
Rationale

6.5 An AES Example

Results
Avalanche Effect

6.6 AES Implementation

Equivalent Inverse Cipher
Implementation Aspects

6.7 Key Terms, Review Questions, and Problems

Appendix 6A Polynomials with Coefficients in GF(28)

CHAPTER

Advanced Encryption Standard

172 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

The Advanced Encryption Standard (AES) was published by the National Institute of
Standards and Technology (NIST) in 2001. AES is a symmetric block cipher that is
intended to replace DES as the approved standard for a wide range of applications.
Compared to public-key ciphers such as RSA, the structure of AES and most symmet-
ric ciphers is quite complex and cannot be explained as easily as many other
cryptographic algorithms. Accordingly, the reader may wish to begin with a simplified
version of AES, which is described in Appendix I. This version allows the reader to
perform encryption and decryption by hand and gain a good understanding of the
working of the algorithm details. Classroom experience indicates that a study of this
simplified version enhances understanding of AES.1 One possible approach is to read
the chapter first, then carefully read Appendix I, and then re-read the main body
of the chapter.

Appendix H looks at the evaluation criteria used by NIST to select from among
the candidates for AES, plus the rationale for picking Rijndael, which was the winning
candidate. This material is useful in understanding not just the AES design but also the
criteria by which to judge any symmetric encryption algorithm.

6.1 FINITE FIELD ARITHMETIC

In AES, all operations are performed on 8-bit bytes. In particular, the arithmetic
operations of addition, multiplication, and division are performed over the finite
field GF(28). Section 5.6 discusses such operations in some detail. For the reader
who has not studied Chapter 5, and as a quick review for those who have, this sec-
tion summarizes the important concepts.

In essence, a field is a set in which we can do addition, subtraction, multiplica-
tion, and division without leaving the set. Division is defined with the following rule:
a/b = a(b-1). An example of a finite field (one with a finite number of elements) is
the set Zp consisting of all the integers {0, 1, c , p - 1}, where p is a prime num-
ber and in which arithmetic is carried out modulo p.

1However, you may safely skip Appendix I, at least on a first reading. If you get lost or bogged down in
the details of AES, then you can go back and start with simplified AES.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Present an overview of the general structure of Advanced Encryption
Standard (AES).

◆ Understand the four transformations used in AES.

◆ Explain the AES key expansion algorithm.

◆ Understand the use of polynomials with coefficients in GF(28).

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6.1 / FINITE FIELD ARITHMETIC 173

Virtually all encryption algorithms, both conventional and public-key, involve
arithmetic operations on integers. If one of the operations used in the algorithm
is division, then we need to work in arithmetic defined over a field; this is because
division requires that each nonzero element have a multiplicative inverse. For con-
venience and for implementation efficiency, we would also like to work with inte-
gers that fit exactly into a given number of bits, with no wasted bit patterns. That is,
we wish to work with integers in the range 0 through 2n - 1, which fit into an n-bit
word. Unfortunately, the set of such integers, Z2n, using modular arithmetic, is not a
field. For example, the integer 2 has no multiplicative inverse in Z2n, that is, there is
no integer b, such that 2b mod 2n = 1.

There is a way of defining a finite field containing 2n elements; such a field is
referred to as GF(2n). Consider the set, S, of all polynomials of degree n - 1 or less
with binary coefficients. Thus, each polynomial has the form

f(x) = an - 1xn - 1 + an - 2xn - 2 + g + a1x + a0 = a
n - 1

i=0
aix

i

where each ai takes on the value 0 or 1. There are a total of 2
n different polynomials

in S. For n = 3, the 23 = 8 polynomials in the set are

0 x x2 x2 + x
1 x + 1 x2 + 1 x2 + x + 1

With the appropriate definition of arithmetic operations, each such set S is a
finite field. The definition consists of the following elements.

1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic
rules of algebra with the following two refinements.

2. Arithmetic on the coefficients is performed modulo 2. This is the same as the
XOR operation.

3. If multiplication results in a polynomial of degree greater than n - 1, then the
polynomial is reduced modulo some irreducible polynomial m(x) of degree n.
That is, we divide by m(x) and keep the remainder. For a polynomial f(x),
the remainder is expressed as r(x) = f(x) mod m(x). A polynomial m(x) is
called irreducible if and only if m(x) cannot be expressed as a product of two
polynomials, both of degree lower than that of m(x).

For example, to construct the finite field GF(23), we need to choose an irre-
ducible polynomial of degree 3. There are only two such polynomials: (x3 + x2 + 1)
and (x3 + x + 1). Addition is equivalent to taking the XOR of like terms. Thus,
(x + 1) + x = 1.

A polynomial in GF(2n) can be uniquely represented by its n binary coeffi cients
(an - 1an - 2 c a0). Therefore, every polynomial in GF(2n) can be represented by
an n-bit number. Addition is performed by taking the bitwise XOR of the two n-bit
elements. There is no simple XOR operation that will accomplish multiplication in
GF(2n). However, a reasonably straightforward, easily implemented, technique is
available. In essence, it can be shown that multiplication of a number in GF(2n) by

174 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

2 consists of a left shift followed by a conditional XOR with a constant. Multiplication
by larger numbers can be achieved by repeated application of this rule.

For example, AES uses arithmetic in the finite field GF(28) with the irreducible
polynomial m(x) = x8 + x4 + x3 + x + 1. Consider two elements A =
(a7a6 c a1a0) and B = (b7b6 c b1b0). The sum A + B = (c7c6 c c1c0), where
ci = ai⊕ bi. The multiplication {02} # A equals (a6 c a1a00) if a7 = 0 and equals
(a6 c a1a00)⊕ (00011011) if a7 = 1.2

To summarize, AES operates on 8-bit bytes. Addition of two bytes is defined
as the bitwise XOR operation. Multiplication of two bytes is defined as multiplica-
tion in the finite field GF(28), with the irreducible polynomial3 m(x) = x8 + x4 + x3 +
x + 1. The developers of Rijndael give as their motivation for selecting this one of
the 30 possible irreducible polynomials of degree 8 that it is the first one on the list
given in [LIDL94].

6.2 AES STRUCTURE

General Structure

Figure 6.1 shows the overall structure of the AES encryption process. The cipher
takes a plaintext block size of 128 bits, or 16 bytes. The key length can be 16, 24, or
32 bytes (128, 192, or 256 bits). The algorithm is referred to as AES-128, AES-192,
or AES-256, depending on the key length.

The input to the encryption and decryption algorithms is a single 128-bit block.
In FIPS PUB 197, this block is depicted as a 4 * 4 square matrix of bytes. This
block is copied into the State array, which is modified at each stage of encryption or
decryption. After the final stage, State is copied to an output matrix. These opera-
tions are depicted in Figure 6.2a. Similarly, the key is depicted as a square matrix of
bytes. This key is then expanded into an array of key schedule words. Figure 6.2b
shows the expansion for the 128-bit key. Each word is four bytes, and the total key
schedule is 44 words for the 128-bit key. Note that the ordering of bytes within a ma-
trix is by column. So, for example, the first four bytes of a 128-bit plaintext input to
the encryption cipher occupy the first column of the in matrix, the second four bytes
occupy the second column, and so on. Similarly, the first four bytes of the expanded
key, which form a word, occupy the first column of the w matrix.

The cipher consists of N rounds, where the number of rounds depends on the
key length: 10 rounds for a 16-byte key, 12 rounds for a 24-byte key, and 14 rounds
for a 32-byte key (Table 6.1). The first N - 1 rounds consist of four distinct trans-
formation functions: SubBytes, ShiftRows, MixColumns, and AddRoundKey,
which are described subsequently. The final round contains only three transforma-
tions, and there is a initial single transformation (AddRoundKey) before the first
round, which can be considered Round 0. Each transformation takes one or more

2In FIPS PUB 197, a hexadecimal number is indicated by enclosing it in curly brackets. We use that convention
in this chapter.
3In the remainder of this discussion, references to GF(28) refer to the finite field defined with this
polynomial.

6.2 / AES STRUCTURE 175

Figure 6.1 AES Encryption Process

Initial transformation

K
ey

e
xp

an
si

on

Plaintext—16 bytes (128 bits) Key—M bytes

Key
(M bytes)Round 0 key

(16 bytes)

Round 1 key
(16 bytes)

Round N – 1 key
(16 bytes)

Round N key
(16 bytes)

Cipehertext—16 bytes (128 bits)

No. of
rounds

10 16

Key
Length
(bytes)

Input state
(16 bytes)

State after
initial

transformation
(16 bytes)

Final state
(16 bytes)

Round N – 1
output state
(16 bytes)

Round 1
output state
(16 bytes)

Round 1
(4 transformations)

Round N – 1
(4 transformations)

Round N
(3 transformations)

12 24

14 32

4 * 4 matrices as input and produces a 4 * 4 matrix as output. Figure 6.1 shows
that the output of each round is a 4 * 4 matrix, with the output of the final round
being the ciphertext. Also, the key expansion function generates N + 1 round keys,
each of which is a distinct 4 * 4 matrix. Each round key serves as one of the inputs
to the AddRoundKey transformation in each round.

F
ig

ur
e

6.
2

A
E

S
D

at
a

St
ru

ct
ur

es

in
0

in
4

in
8

in
12

in
1

in
5

in
9

in
13

in
2

in
6

in
10

in
14

in
3

in
7

in
11

in
15

k 0

w
0

w
1

w
2

w
43

w
42

k 4
k 8

k 1
2

k 1
k 5

k 9
k 1

3

k 2
k 6

k 1
0

k 1
4

k 3
k 7

k 1
1

k 1
5

ou
t 0

ou
t 4

ou
t 8

ou
t 1

2

ou
t 1

ou
t 5

ou
t 9

ou
t 1

3

ou
t 2

ou
t 6

ou
t 1

0
ou

t 1
4

ou
t 3

ou
t 7

ou
t 1

1
ou

t 1
5

s 0
,0

s 1
,0

s 2
,0

s 3
,0

s 0
,1

s 1
,1

s 2
,1

s 3
,1

s 0
,2

s 1
,2

s 2
,2

s 3
,2

s 0
,3

s 1
,3

s 2
,3

s 3
,3

s 0
,0

s 1
,0

s 2
,0

s 3
,0

s 0
,1

s 1
,1

s 2
,1

s 3
,1

s 0
,2

s 1
,2

s 2
,2

s 3
,2

s 0
,3

s 1
,3

s 2
,3

s 3
,3

(a
) I

np
ut

, s
ta

te
a

rr
ay

, a
nd

o
ut

pu
t

(b
) K

ey
a

nd
e

xp
an

de
d

ke
y

176

6.2 / AES STRUCTURE 177

Key Size (words/bytes/bits) 4/16/128 6/24/192 8/32/256
Plaintext Block Size (words/bytes/bits) 4/16/128 4/16/128 4/16/128
Number of Rounds 10 12 14
Round Key Size (words/bytes/bits) 4/16/128 4/16/128 4/16/128
Expanded Key Size (words/bytes) 44/176 52/208 60/240

Table 6.1 AES Parameters

Detailed Structure

Figure 6.3 shows the AES cipher in more detail, indicating the sequence of transfor-
mations in each round and showing the corresponding decryption function. As was
done in Chapter 4, we show encryption proceeding down the page and decryption
proceeding up the page.

Before delving into details, we can make several comments about the overall
AES structure.

1. One noteworthy feature of this structure is that it is not a Feistel structure.
Recall that, in the classic Feistel structure, half of the data block is used to
modify the other half of the data block and then the halves are swapped. AES
instead processes the entire data block as a single matrix during each round
using substitutions and permutation.

2. The key that is provided as input is expanded into an array of forty-four 32-bit
words, w[i]. Four distinct words (128 bits) serve as a round key for each round;
these are indicated in Figure 6.3.

3. Four different stages are used, one of permutation and three of substitution:

■ Substitute bytes: Uses an S-box to perform a byte-by-byte substitution of
the block.

■ ShiftRows: A simple permutation.

■ MixColumns: A substitution that makes use of arithmetic over GF(28).

■ AddRoundKey: A simple bitwise XOR of the current block with a portion
of the expanded key.

4. The structure is quite simple. For both encryption and decryption, the cipher
begins with an AddRoundKey stage, followed by nine rounds that each in-
cludes all four stages, followed by a tenth round of three stages. Figure 6.4
depicts the structure of a full encryption round.

5. Only the AddRoundKey stage makes use of the key. For this reason, the cipher
begins and ends with an AddRoundKey stage. Any other stage, applied at the
beginning or end, is reversible without knowledge of the key and so would add
no security.

6. The AddRoundKey stage is, in effect, a form of Vernam cipher and by itself
would not be formidable. The other three stages together provide confusion,
diffusion, and nonlinearity, but by themselves would provide no security be-
cause they do not use the key. We can view the cipher as alternating operations
of XOR encryption (AddRoundKey) of a block, followed by scrambling of the

178 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Figure 6.3 AES Encryption and Decryption

Add round key

w[4, 7]

Plaintext
(16 bytes)

Plaintext
(16 bytes)

Substitute bytes

Expand key

Shift rows

Mix columnsR
ou

nd
1

R
ou

nd
9

R
ou

nd
1

0

Add round key

Substitute bytes

Shift rows

Mix columns

Add round key

Substitute bytes

Shift rows

Add round key

Ciphertext
(16 bytes)

(a) Encryption

Key
(16 bytes)

Add round key

Inverse sub bytes

Inverse shift rows

Inverse mix cols

R
ou

nd
1

0
R

ou
nd

9
R

ou
nd

1

Add round key

Inverse sub bytes

Inverse shift rows

Inverse mix cols

Add round key

Inverse sub bytes

Inverse shift rows

Add round key

Ciphertext
(16 bytes)

(b) Decryption

w[36, 39]

w[40, 43]

w[0, 3]

block (the other three stages), followed by XOR encryption, and so on. This
scheme is both efficient and highly secure.

7. Each stage is easily reversible. For the Substitute Byte, ShiftRows, and
MixColumns stages, an inverse function is used in the decryption algorithm.
For the AddRoundKey stage, the inverse is achieved by XORing the same
round key to the block, using the result that A⊕ B⊕ B = A.

8. As with most block ciphers, the decryption algorithm makes use of the
expanded key in reverse order. However, the decryption algorithm is not

6.3 / AES TRANSFORMATION FUNCTIONS 179

Figure 6.4 AES Encryption Round

SSubBytes

State

State

State

State

State

ShiftRows

MixColumns

AddRoundKey

S S S S S S S S S S S S S S S

M M M M

r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15

identical to the encryption algorithm. This is a consequence of the particular
structure of AES.

9. Once it is established that all four stages are reversible, it is easy to verify
that decryption does recover the plaintext. Figure 6.3 lays out encryption
and decryption going in opposite vertical directions. At each horizontal point
(e.g., the dashed line in the figure), State is the same for both encryption and
decryption.

10. The final round of both encryption and decryption consists of only three stages.
Again, this is a consequence of the particular structure of AES and is required
to make the cipher reversible.

6.3 AES TRANSFORMATION FUNCTIONS

We now turn to a discussion of each of the four transformations used in AES. For
each stage, we describe the forward (encryption) algorithm, the inverse ( decryption)
algorithm, and the rationale for the stage.

180 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Substitute Bytes Transformation

FORWARD AND INVERSE TRANSFORMATIONS The forward substitute byte
transformation, called SubBytes, is a simple table lookup (Figure 6.5a). AES
defines a 16 * 16 matrix of byte values, called an S-box (Table 6.2a), that con-
tains a permutation of all possible 256 8-bit values. Each individual byte of State
is mapped into a new byte in the following way: The leftmost 4 bits of the byte are
used as a row value and the rightmost 4 bits are used as a column value. These row
and column values serve as indexes into the S-box to select a unique 8-bit output
value. For example, the hexadecimal value {95} references row 9, column 5 of the
S-box, which contains the value {2A}. Accordingly, the value {95} is mapped into
the value {2A}.

Figure 6.5 AES Byte-Level Operations

s0,0 s0,1 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

(b) Add round key transformation

(a) Substitute byte transformation

S-box

x

y

¿ ¿ ¿ ¿

¿ ¿¿¿

s1,1

s0,0

wi wi+2 wi+3

s0,2 s0,3

s1,0 s1,2 s1,3
=

s2,0 s2,2 s2,3

s3,0 s3,2 s3,3

s1,1

s0,0 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,2 s2,3

s3,0 s3,2 s3,3

s1,1

s0,1

s2,1

s3,1

wi+1

s0,1

s2,1

s3,1

s1,1

¿¿¿

¿ ¿ ¿ ¿

¿

¿

¿

¿
¿

¿ ¿

¿ ¿ ¿ ¿

¿ ¿ ¿

¿
¿ ¿

6.3 / AES TRANSFORMATION FUNCTIONS 181

y
0 1 2 3 4 5 6 7 8 9 A B C D E F

0 63 7C 77 7B F2 6B 6F C5 30 01 67 2B FE D7 AB 76
1 CA 82 C9 7D FA 59 47 F0 AD D4 A2 AF 9C A4 72 C0
2 B7 FD 93 26 36 3F F7 CC 34 A5 E5 F1 71 D8 31 15
3 04 C7 23 C3 18 96 05 9A 07 12 80 E2 EB 27 B2 75
4 09 83 2C 1A 1B 6E 5A A0 52 3B D6 B3 29 E3 2F 84
5 53 D1 00 ED 20 FC B1 5B 6A CB BE 39 4A 4C 58 CF
6 D0 EF AA FB 43 4D 33 85 45 F9 02 7F 50 3C 9F A8

x
7 51 A3 40 8F 92 9D 38 F5 BC B6 DA 21 10 FF F3 D2
8 CD 0C 13 EC 5F 97 44 17 C4 A7 7E 3D 64 5D 19 73
9 60 81 4F DC 22 2A 90 88 46 EE B8 14 DE 5E 0B DB
A E0 32 3A 0A 49 06 24 5C C2 D3 AC 62 91 95 E4 79
B E7 C8 37 6D 8D D5 4E A9 6C 56 F4 EA 65 7A AE 08
C BA 78 25 2E 1C A6 B4 C6 E8 DD 74 1F 4B BD 8B 8A
D 70 3E B5 66 48 03 F6 0E 61 35 57 B9 86 C1 1D 9E
E E1 F8 98 11 69 D9 8E 94 9B 1E 87 E9 CE 55 28 DF
F 8C A1 89 0D BF E6 42 68 41 99 2D 0F B0 54 BB 16

(a) S-box

y
0 1 2 3 4 5 6 7 8 9 A B C D E F

0 52 09 6A D5 30 36 A5 38 BF 40 A3 9E 81 F3 D7 FB
1 7C E3 39 82 9B 2F FF 87 34 8E 43 44 C4 DE E9 CB
2 54 7B 94 32 A6 C2 23 3D EE 4C 95 0B 42 FA C3 4E
3 08 2E A1 66 28 D9 24 B2 76 5B A2 49 6D 8B D1 25
4 72 F8 F6 64 86 68 98 16 D4 A4 5C CC 5D 65 B6 92
5 6C 70 48 50 FD ED B9 DA 5E 15 46 57 A7 8D 9D 84
6 90 D8 AB 00 8C BC D3 0A F7 E4 58 05 B8 B3 45 06

x
7 D0 2C 1E 8F CA 3F 0F 02 C1 AF BD 03 01 13 8A 6B
8 3A 91 11 41 4F 67 DC EA 97 F2 CF CE F0 B4 E6 73
9 96 AC 74 22 E7 AD 35 85 E2 F9 37 E8 1C 75 DF 6E
A 47 F1 1A 71 1D 29 C5 89 6F B7 62 0E AA 18 BE 1B
B FC 56 3E 4B C6 D2 79 20 9A DB C0 FE 78 CD 5A F4
C 1F DD A8 33 88 07 C7 31 B1 12 10 59 27 80 EC 5F
D 60 51 7F A9 19 B5 4A 0D 2D E5 7A 9F 93 C9 9C EF
E A0 E0 3B 4D AE 2A F5 B0 C8 EB BB 3C 83 53 99 61
F 17 2B 04 7E BA 77 D6 26 E1 69 14 63 55 21 0C 7D

(b) Inverse S-box

Table 6.2 AES S-Boxes

Hiva-Network.Com

182 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Here is an example of the SubBytes transformation:

EA 04 65 85 87 F2 4D 97

83 45 5D 96 EC 6E 4C 90

5C 33 98 B0 S 4A C3 46 E7
F0 2D AD C5 8C D8 95 A6

The S-box is constructed in the following fashion (Figure 6.6a).

Figure 6.6 Constuction of S-Box and IS-Box

b0
b1
b2
b3
b4
b5
b6
b7

=

1 0 0 0 1 1 1 1
1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1
1 1 1 1 0 0 0 1
1 1 1 1 1 0 0 0
0 1 1 1 1 1 0 0
0 0 1 1 1 1 1 0
0 0 0 1 1 1 1 1

b0
b1
b2
b3
b4
b5
b6
b7

+

1
1
0
0
0
1
1
0

Inverse
in GF(28)

Byte to bit
column vector

Bit column
vector to byte

Byte at row y,
column x

initialized to yx
yx

S(yx)

(a) Calculation of byte at
row y, column x of S-box

(a) Calculation of byte at
row y, column x of IS-box

Inverse
in GF(28)

Byte to bit
column vector

Bit column
vector to byte

Byte at row y,
column x

initialized to yx
yx

b0¿

b¿

b¿
b ¿

1

2

3

b4
b5
b6
b7

=

0 0 1 0 0 1 0 1
1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 1
1 0 1 0 0 1 0 0
0 1 0 1 0 0 1 0
0 0 1 0 1 0 0 1
1 0 0 1 0 1 0 0
0 1 0 0 1 0 1 0

b0
b1
b2
b3
b4
b5
b6
b7

+

1
0
1
0
0
0
0
0

IS(yx)

¿

¿
¿

¿

¿

¿

¿

¿

¿

¿

¿
¿

6.3 / AES TRANSFORMATION FUNCTIONS 183

1. Initialize the S-box with the byte values in ascending sequence row by row.
The first row contains {00}, {01}, {02}, c , {0F}; the second row contains
{10}, {11}, etc.; and so on. Thus, the value of the byte at row y, column x is {yx}.

2. Map each byte in the S-box to its multiplicative inverse in the finite field
GF(28); the value {00} is mapped to itself.

3. Consider that each byte in the S-box consists of 8 bits labeled
(b7, b6, b5, b4, b3, b2, b1, b0). Apply the following transformation to each bit of
each byte in the S-box:

bi
= = bi⊕ b(i+ 4) mod 8 ⊕ b(i+ 5) mod 8 ⊕ b(i+ 6) mod 8 ⊕ b(i+ 7) mod 8 ⊕ ci (6.1)

where ci is the ith bit of byte c with the value {63}; that is,
(c7c6c5c4c3c2c1c0) = (01100011). The prime (′) indicates that the variable is to
be updated by the value on the right. The AES standard depicts this transfor-
mation in matrix form as follows.

H
b0
=

b1
=

b2
=

b3
=

b4
=

b5
=

b6
=

b7
=

X = H
1 0 0 0 1 1 1 1
1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1
1 1 1 1 0 0 0 1
1 1 1 1 1 0 0 0
0 1 1 1 1 1 0 0
0 0 1 1 1 1 1 0
0 0 0 1 1 1 1 1

X H
b0
b1
b2
b3
b4
b5
b6
b7

X + H
1
1
0
0
0
1
1
0

X (6.2)
Equation (6.2) has to be interpreted carefully. In ordinary matrix multiplica-

tion,4 each element in the product matrix is the sum of products of the elements of
one row and one column. In this case, each element in the product matrix is the
bitwise XOR of products of elements of one row and one column. Furthermore, the
final addition shown in Equation (6.2) is a bitwise XOR. Recall from Section 5.6
that the bitwise XOR is addition in GF(28).

As an example, consider the input value {95}. The multiplicative inverse in
GF(28) is {95}-1 = {8A}, which is 10001010 in binary. Using Equation (6.2),

H
1 0 0 0 1 1 1 1
1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1
1 1 1 1 0 0 0 1
1 1 1 1 1 0 0 0
0 1 1 1 1 1 0 0
0 0 1 1 1 1 1 0
0 0 0 1 1 1 1 1

X H
0
1
0
1
0
0
0
1

X ⊕ H
1
1
0
0
0
1
1
0

X = H
1
0
0
1
0
0
1
0

X ⊕ H
1
1
0
0
0
1
1
0

X = H
0
1
0
1
0
1
0
0

X

4For a brief review of the rules of matrix and vector multiplication, refer to Appendix E.

184 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

The result is {2A}, which should appear in row {09} column {05} of the S-box.
This is verified by checking Table 6.2a.

The inverse substitute byte transformation, called InvSubBytes, makes use
of the inverse S-box shown in Table 6.2b. Note, for example, that the input {2A}
produces the output {95}, and the input {95} to the S-box produces {2A}. The inverse
S-box is constructed (Figure 6.6b) by applying the inverse of the transformation in
Equation (6.1) followed by taking the multiplicative inverse in GF(28). The inverse
transformation is

bi
= = b(i+ 2) mod 8 ⊕ b(i+ 5) mod 8 ⊕ b(i+ 7) mod 8 ⊕ di

where byte d = {05}, or 00000101. We can depict this transformation as follows.

H
b0
=

b1
=

b2
=

b3
=

b4
=

b5
=

b6
=

b7
=

X = H
0 0 1 0 0 1 0 1
1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 1
1 0 1 0 0 1 0 0
0 1 0 1 0 0 1 0
0 0 1 0 1 0 0 1
1 0 0 1 0 1 0 0
0 1 0 0 1 0 1 0

X H
b0
b1
b2
b3
b4
b5
b6
b7

X + H
1
0
1
0
0
0
0
0

X
To see that InvSubBytes is the inverse of SubBytes, label the matrices in

SubBytes and InvSubBytes as X and Y, respectively, and the vector versions of con-
stants c and d as C and D, respectively. For some 8-bit vector B, Equation (6.2)
becomes B= = XB⊕ C. We need to show that Y(XB⊕ C)⊕D = B. To multiply
out, we must show YXB⊕ YC⊕D = B. This becomes

H
0 0 1 0 0 1 0 1
1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 1
1 0 1 0 0 1 0 0
0 1 0 1 0 0 1 0
0 0 1 0 1 0 0 1
1 0 0 1 0 1 0 0
0 1 0 0 1 0 1 0

X H
1 0 0 0 1 1 1 1
1 1 0 0 0 1 1 1
1 1 1 0 0 0 1 1
1 1 1 1 0 0 0 1
1 1 1 1 1 0 0 0
0 1 1 1 1 1 0 0
0 0 1 1 1 1 1 0
0 0 0 1 1 1 1 1

X H
b0
b1
b2
b3
b4
b5
b6
b7

X ⊕

H
0 0 1 0 0 1 0 1
1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 1
1 0 1 0 0 1 0 0
0 1 0 1 0 0 1 0
0 0 1 0 1 0 0 1
1 0 0 1 0 1 0 0
0 1 0 0 1 0 1 0

X H
1
1
0
0
0
1
1
0

X ⊕ H
1
0
1
0
0
0
0
0

X =

6.3 / AES TRANSFORMATION FUNCTIONS 185

H
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1

X H
b0
b1
b2
b3
b4
b5
b6
b7

X ⊕ H
1
0
1
0
0
0
0
0

X ⊕ H
1
0
1
0
0
0
0
0

X = H
b0
b1
b2
b3
b4
b5
b6
b7

X
We have demonstrated that YX equals the identity matrix, and the YC = D,

so that YC⊕D equals the null vector.

RATIONALE The S-box is designed to be resistant to known cryptanalytic attacks.
Specifically, the Rijndael developers sought a design that has a low correlation
between input bits and output bits and the property that the output is not a linear
mathematical function of the input [DAEM01]. The nonlinearity is due to the use
of the multiplicative inverse. In addition, the constant in Equation (6.1) was chosen
so that the S-box has no fixed points [[email protected](a) = a] and no “opposite fixed points”
[[email protected](a) = a], where a is the bitwise complement of a.

Of course, the S-box must be invertible, that is, [email protected][[email protected](a)] = a.
However, the S-box does not self-inverse in the sense that it is not true that
[email protected](a) = [email protected](a). For example, [email protected]({95}) = {2A}, but [email protected]({95}) = {AD}.

ShiftRows Transformation

FORWARD AND INVERSE TRANSFORMATIONS The forward shift row transformation,
called ShiftRows, is depicted in Figure 6.7a. The first row of State is not altered. For
the second row, a 1-byte circular left shift is performed. For the third row, a 2-byte
circular left shift is performed. For the fourth row, a 3-byte circular left shift is per-
formed. The following is an example of ShiftRows.

87 F2 4D 97 87 F2 4D 97

EC 6E 4C 90 6E 4C 90 EC

4A C3 46 E7 S 46 E7 4A C3
8C D8 95 A6 A6 8C D8 95

The inverse shift row transformation, called InvShiftRows, performs the cir-
cular shifts in the opposite direction for each of the last three rows, with a 1-byte
circular right shift for the second row, and so on.

RATIONALE The shift row transformation is more substantial than it may first
appear. This is because the State, as well as the cipher input and output, is
treated as an array of four 4-byte columns. Thus, on encryption, the first 4 bytes
of the plaintext are copied to the first column of State, and so on. Furthermore,
as will be seen, the round key is applied to State column by column. Thus, a row
shift moves an individual byte from one column to another, which is a linear

186 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

5We follow the convention of FIPS PUB 197 and use the symbol # to indicate multiplication over the
finite field GF(28) and ⊕ to indicate bitwise XOR, which corresponds to addition in GF(28).

distance of a multiple of 4 bytes. Also note that the transformation ensures that
the 4 bytes of one column are spread out to four different columns. Figure 6.4
illustrates the effect.

MixColumns Transformation

FORWARD AND INVERSE TRANSFORMATIONS The forward mix column transformation,
called MixColumns, operates on each column individually. Each byte of a column
is mapped into a new value that is a function of all four bytes in that column. The
transformation can be defined by the following matrix multiplication on State
(Figure 6.7b):

D02 03 01 0101 02 03 01
01 01 02 03
03 01 01 02

T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s3,0 s3,1 s3,2 s3,3

T = D s0,0= s0,1= s0,2= s0,3=s1,0= s1,1= s1,2= s1,3=
s2,0
= s2,1

= s2,2
= s2,3

=

s3,0
= s3,1

= s3,2
= s3,3

=

T (6.3)
Each element in the product matrix is the sum of products of elements of one row
and one column. In this case, the individual additions and multiplications5 are

Figure 6.7 AES Row and Column Operations

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,1 s1,2 s1,3 s1,0

s2,2 s2,3 s2,0 s2,1

s3,3 s3,0 s3,1 s3,2

(a) Shift row transformation

(b) Mix column transformation

2 3 1 1
1 2 3 1
1 1 2 3
3 1 1 2

=*

¿ ¿ ¿ ¿

¿¿¿¿

¿ ¿ ¿ ¿

¿¿¿¿

6.3 / AES TRANSFORMATION FUNCTIONS 187

performed in GF(28). The MixColumns transformation on a single column of State
can be expressed as

s0, j
= = (2 # s0, j)⊕ (3 # s1, j)⊕ s2, j⊕ s3, j

s1, j
= = s0, j⊕ (2 # s1, j)⊕ (3 # s2, j)⊕ s3, j

s2, j
= = s0, j⊕ s1, j⊕ (2 # s2, j)⊕ (3 # s3, j)

s3, j
= = (3 # s0, j)⊕ s1, j⊕ s2, j⊕ (2 # s3, j)

(6.4)

The following is an example of MixColumns:

87 F2 4D 97 47 40 A3 4C

6E 4C 90 EC 37 D4 70 9F

46 E7 4A C3 S 94 E4 3A 42
A6 8C D8 95 ED A5 A6 BC

Let us verify the first column of this example. Recall from Section 5.6 that, in
GF(28), addition is the bitwise XOR operation and that multiplication can be per-
formed according to the rule established in Equation (4.14). In particular, multipli-
cation of a value by x (i.e., by {02}) can be implemented as a 1-bit left shift followed
by a conditional bitwise XOR with (0001 1011) if the leftmost bit of the original
value (prior to the shift) is 1. Thus, to verify the MixColumns transformation on the
first column, we need to show that

({02} # {87}) ⊕ ({03} # {6E}) ⊕ {46} ⊕ {A6} = {47}
{87} ⊕ ({02} # {6E}) ⊕ ({03} # {46}) ⊕ {A6} = {37}
{87} ⊕ {6E} ⊕ ({02} # {46}) ⊕ ({03} # {A6}) = {94}
({03} # {87}) ⊕ {6E} ⊕ {46} ⊕ ({02} # {A6}) = {ED}
For the first equation, we have {02} # {87} = (0000 1110)⊕ (0001 1011) =

(0001 0101) and {03} # {6E} = {6E}⊕ ({02} # {6E}) = (0110 1110)⊕ (1101 1100) =
(1011 0010). Then,

{02} # {87} = 0001 0101
{03} # {6E} = 1011 0010
{46} = 0100 0110
{A6} = 1010 0110

0100 0111 = {47}

The other equations can be similarly verified.
The inverse mix column transformation, called InvMixColumns, is defined by

the following matrix multiplication:

D 0E 0B 0D 0909 0E 0B 0D
0D 09 0E 0B
0B 0D 09 0E

T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s3,0 s3,1 s3,2 s3,3

T = D s0,0= s0,1= s0,2= s0,3=s1,0= s1,1= s1,2= s1,3=
s2,0
= s2,1

= s2,2
= s2,3

=

s3,0
= s3,1

= s3,2
= s3,3

=

T (6.5)

188 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

It is not immediately clear that Equation (6.5) is the inverse of Equation (6.3).
We need to show

D 0E 0B 0D 0909 0E 0B 0D
0D 09 0E 0B
0B 0D 09 0E

T D02 03 01 0101 02 03 01
01 01 02 03
03 01 01 02

T D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s3,0 s3,1 s3,2 s3,3

T = D s0,0 s0,1 s0,2 s0,3s1,0 s1,1 s1,2 s1,3
s2,0 s2,1 s2,2 s2,3
s0,3 s3,1 s3,2 s3,3

T
which is equivalent to showing

D 0E 0B 0D 0909 0E 0B 0D
0D 09 0E 0B
0B 0D 09 0E

T D02 03 01 0101 02 03 01
01 01 02 03
03 01 01 02

T = D1 0 0 00 1 0 0
0 0 1 0
0 0 0 1

T (6.6)
That is, the inverse transformation matrix times the forward transformation matrix
equals the identity matrix. To verify the first column of Equation (6.6), we need
to show

({0E} # {02})⊕ {0B}⊕ {0D}⊕ ({09} # {03}) = {01}
({09} # {02})⊕ {0E}⊕ {0B}⊕ ({0D} # {03}) = {00}
({0D} # {02})⊕ {09}⊕ {0E}⊕ ({0B} # {03}) = {00}

({0B} # {02})⊕ {0D}⊕ {09}⊕ ({0E} # {03}) = {00}
For the first equation, we have {0E} # {02} = 00011100 and {09} # {03} =

{09}⊕ ({09} # {02}) = 00001001⊕ 00010010 = 00011011. Then

{0E} # {02} = 00011100
{0B} = 00001011
{0D} = 00001101
{09} # {03} = 00011011

00000001

The other equations can be similarly verified.
The AES document describes another way of characterizing the MixColumns

transformation, which is in terms of polynomial arithmetic. In the standard,
MixColumns is defined by considering each column of State to be a four-term poly-
nomial with coefficients in GF(28). Each column is multiplied modulo (x4 + 1) by
the fixed polynomial a(x), given by

a(x) = {03}x3 + {01}x2 + {01}x + {02} (6.7)

Appendix 5A demonstrates that multiplication of each column of State by
a(x) can be written as the matrix multiplication of Equation (6.3). Similarly, it
can be seen that the transformation in Equation (6.5) corresponds to treating

6.3 / AES TRANSFORMATION FUNCTIONS 189

each column as a four-term polynomial and multiplying each column by b(x),
given by

b(x) = {0B}x3 + {0D}x2 + {09}x + {0E} (6.8)

It readily can be shown that b(x) = a-1(x) mod (x4 + 1).

RATIONALE The coefficients of the matrix in Equation (6.3) are based on a linear
code with maximal distance between code words, which ensures a good mixing
among the bytes of each column. The mix column transformation combined with
the shift row transformation ensures that after a few rounds all output bits depend
on all input bits. See [DAEM99] for a discussion.

In addition, the choice of coefficients in MixColumns, which are all {01}, {02},
or {03}, was influenced by implementation considerations. As was discussed, multi-
plication by these coefficients involves at most a shift and an XOR. The coefficients
in InvMixColumns are more formidable to implement. However, encryption was
deemed more important than decryption for two reasons:

1. For the CFB and OFB cipher modes (Figures 7.5 and 7.6; described in
Chapter 7), only encryption is used.

2. As with any block cipher, AES can be used to construct a message authentica-
tion code (Chapter 13), and for this, only encryption is used.

AddRoundKey Transformation

FORWARD AND INVERSE TRANSFORMATIONS In the forward add round key transfor-
mation, called AddRoundKey, the 128 bits of State are bitwise XORed with the
128 bits of the round key. As shown in Figure 6.5b, the operation is viewed as a
columnwise operation between the 4 bytes of a State column and one word of
the round key; it can also be viewed as a byte-level operation. The following is an
example of AddRoundKey:

47 40 A3 4C AC 19 28 57 EB 59 8B 1B

37 D4 70 9F 77 FA D1 5C 40 2E A1 C3

94 E4 3A 42 ⊕ 66 DC 29 00 = F2 38 13 42

ED A5 A6 BC F3 21 41 6A 1E 84 E7 D6

The first matrix is State, and the second matrix is the round key.
The inverse add round key transformation is identical to the forward add

round key transformation, because the XOR operation is its own inverse.

RATIONALE The add round key transformation is as simple as possible and affects
every bit of State. The complexity of the round key expansion, plus the complexity
of the other stages of AES, ensure security.

Figure 6.8 is another view of a single round of AES, emphasizing the mecha-
nisms and inputs of each transformation.

190 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

6.4 AES KEY EXPANSION

Key Expansion Algorithm

The AES key expansion algorithm takes as input a four-word (16-byte) key and
produces a linear array of 44 words (176 bytes). This is sufficient to provide a four-
word round key for the initial AddRoundKey stage and each of the 10 rounds of the
cipher. The pseudocode on the next page describes the expansion.

The key is copied into the first four words of the expanded key. The remain-
der of the expanded key is filled in four words at a time. Each added word w[i]
depends on the immediately preceding word, w[i - 1], and the word four positions
back, w[i - 4]. In three out of four cases, a simple XOR is used. For a word whose
position in the w array is a multiple of 4, a more complex function is used. Figure 6.9
illustrates the generation of the expanded key, using the symbol g to represent that
complex function. The function g consists of the following subfunctions.

Figure 6.8 Inputs for Single AES Round

SubBytes

State matrix
at beginning

of round

State matrix
at end

of round

MixColumns matrix
Round

key

Variable inputConstant inputs

ShiftRows

MixColumns

AddRoundKey

S-box

02 03 01 01
01 02 03 01
01 01 02 03
03 01 01 02

Hiva-Network.Com

6.4 / AES KEY EXPANSION 191

KeyExpansion (byte key[16], word w[44])
{
word temp
for (i = 0; i < 4; i++) w[i] = (key[4*i], key[4*i+1],
key[4*i+2],
key[4*i+3]);
for (i = 4; i < 44; i++)
{
temp = w[i − 1];
if (i mod 4 = 0) temp = SubWord (RotWord (temp))
⊕ Rcon[i/4];
w[i] = w[i−4] ⊕ temp
}
}

Figure 6.9 AES Key Expansion

k3

(a) Overall algorithm

(b) Function g

k7 k11 k15

k2 k6 k10 k14

k1 k5 k9 k13

k0 k4 k8 k12

w0 w1 w2 w3 g

w4 w5 w6 w7

w40 w41 w42 w43

g

B0 B1 B2 B3

w

w

B1 B2 B3 B0

0 0 0

B1

S S

B2' ' B3

S S

B0' '

RCj

œ

192 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

1. RotWord performs a one-byte circular left shift on a word. This means that an
input word [B0, B1, B2, B3] is transformed into [B1, B2, B3, B0].

2. SubWord performs a byte substitution on each byte of its input word, using the
S-box (Table 6.2a).

3. The result of steps 1 and 2 is XORed with a round constant, Rcon[j].

The round constant is a word in which the three rightmost bytes are always 0.
Thus, the effect of an XOR of a word with Rcon is to only perform an XOR on the
leftmost byte of the word. The round constant is different for each round and is de-
fined as Rcon[j] = (RC[j], 0, 0, 0), with RC[1] = 1, RC[j] = 2 # RC[j - 1] and with
multiplication defined over the field GF(28). The values of RC[j] in hexadecimal are

j 1 2 3 4 5 6 7 8 9 10

RC[j] 01 02 04 08 10 20 40 80 1B 36

For example, suppose that the round key for round 8 is

EA D2 73 21 B5 8D BA D2 31 2B F5 60 7F 8D 29 2F

Then the first 4 bytes (first column) of the round key for round 9 are calculated as
follows:

i (decimal) temp
After

RotWord
After

SubWord
Rcon (9)

After XOR
with Rcon

w[i - 4] w[i] = temp⊕ w[i - 4]
36 7F8D292F 8D292F7F 5DA515D2 1B000000 46A515D2 EAD27321 AC7766F3

Rationale

The Rijndael developers designed the expansion key algorithm to be resistant to
known cryptanalytic attacks. The inclusion of a round-dependent round constant
eliminates the symmetry, or similarity, between the ways in which round keys are
generated in different rounds. The specific criteria that were used are [DAEM99]

■ Knowledge of a part of the cipher key or round key does not enable calcula-
tion of many other round-key bits.

■ An invertible transformation [i.e., knowledge of any Nk consecutive words of
the expanded key enables regeneration of the entire expanded key (Nk = key
size in words)].

■ Speed on a wide range of processors.

■ Usage of round constants to eliminate symmetries.

■ Diffusion of cipher key differences into the round keys; that is, each key bit
affects many round key bits.

■ Enough nonlinearity to prohibit the full determination of round key differ-
ences from cipher key differences only.

■ Simplicity of description.

6.5 / AN AES EXAMPLE 193

The authors do not quantify the first point on the preceding list, but the idea
is that if you know less than Nk consecutive words of either the cipher key or one of
the round keys, then it is difficult to reconstruct the remaining unknown bits. The
fewer bits one knows, the more difficult it is to do the reconstruction or to deter-
mine other bits in the key expansion.

6.5 AN AES EXAMPLE

We now work through an example and consider some of its implications. Although
you are not expected to duplicate the example by hand, you will find it informative
to study the hex patterns that occur from one step to the next.

For this example, the plaintext is a hexadecimal palindrome. The plaintext,
key, and resulting ciphertext are

Plaintext: 0123456789abcdeffedcba9876543210

Key: 0f1571c947d9e8590cb7add6af7f6798

Ciphertext: ff0b844a0853bf7c6934ab4364148fb9

Results

Table 6.3 shows the expansion of the 16-byte key into 10 round keys. As previ-
ously explained, this process is performed word by word, with each four-byte word
occupying one column of the word round-key matrix. The left-hand column shows

Key Words Auxiliary Function

w0 = 0f 15 71 c9
w1 = 47 d9 e8 59
w2 = 0c b7 ad d6
w3 = af 7f 67 98

RotWord (w3) = 7f 67 98 af = x1
SubWord (x1) = d2 85 46 79 = y1
Rcon (1) = 01 00 00 00
y1 ⊕ Rcon (1) = d3 85 46 79 = z1

w4 = w0 ⊕ z1 = dc 90 37 b0
w5 = w4 ⊕ w1 = 9b 49 df e9
w6 = w5 ⊕ w2 = 97 fe 72 3f
w7 = w6 ⊕ w3 = 38 81 15 a7

RotWord (w7) = 81 15 a7 38 = x2
SubWord (x2) = 0c 59 5c 07 = y2
Rcon (2) = 02 00 00 00
y2 ⊕ Rcon (2) = 0e 59 5c 07 = z2

w8 = w4 ⊕ z2 = d2 c9 6b b7
w9 = w8 ⊕ w5 = 49 80 b4 5e
w10 = w9 ⊕ w6 = de 7e c6 61
w11 = w10 ⊕ w7 = e6 ff d3 c6

RotWord (w11) = ff d3 c6 e6 = x3
SubWord (x3) = 16 66 b4 83 = y3
Rcon (3) = 04 00 00 00
y3 ⊕ Rcon (3) = 12 66 b4 8e = z3

w12 = w8 ⊕ z3 = c0 af df 39
w13 = w12 ⊕ w9 = 89 2f 6b 67
w14 = w13 ⊕ w10 = 57 51 ad 06
w15 = w14 ⊕ w11 = b1 ae 7e c0

RotWord (w15) = ae 7e c0 b1 = x4
SubWord (x4) = e4 f3 ba c8 = y4
Rcon (4) = 08 00 00 00
y4 ⊕ Rcon (4) = ec f3 ba c8 = 4

Table 6.3 Key Expansion for AES Example

(Continued)

194 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Key Words Auxiliary Function

w16 = w12 ⊕ z4 = 2c 5c 65 f1
w17 = w16 ⊕ w13 = a5 73 0e 96
w18 = w17 ⊕ w14 = f2 22 a3 90
w19 = w18 ⊕ w15 = 43 8c dd 50

RotWord (w19) = 8c dd 50 43 = x5
SubWord (x5) = 64 c1 53 1a = y5
Rcon(5) = 10 00 00 00
y5 ⊕ Rcon (5) = 74 c1 53 1a = z5

w20 = w16 ⊕ z5 = 58 9d 36 eb
w21 = w20 ⊕ w17 = fd ee 38 7d
w22 = w21 ⊕ w18 = 0f cc 9b ed
w23 = w22 ⊕ w19 = 4c 40 46 bd

RotWord (w23) = 40 46 bd 4c = x6
SubWord (x6) = 09 5a 7a 29 = y6
Rcon(6) = 20 00 00 00
y6 ⊕ Rcon(6) = 29 5a 7a 29 = z6

w24 = w20 ⊕ z6 = 71 c7 4c c2
w25 = w24 ⊕ w21 = 8c 29 74 bf
w26 = w25 ⊕ w22 = 83 e5 ef 52
w27 = w26 ⊕ w23 = cf a5 a9 ef

RotWord (w27) = a5 a9 ef cf = x7
SubWord (x7) = 06 d3 bf 8a = y7
Rcon (7) = 40 00 00 00
y7 ⊕ Rcon(7) = 46 d3 df 8a = z7

w28 = w24 ⊕ z7 = 37 14 93 48
w29 = w28 ⊕ w25 = bb 3d e7 f7
w30 = w29 ⊕ w26 = 38 d8 08 a5
w31 = w30 ⊕ w27 = f7 7d a1 4a

RotWord (w31) = 7d a1 4a f7 = x8
SubWord (x8) = ff 32 d6 68 = y8
Rcon (8) = 80 00 00 00
y8 ⊕ Rcon(8) = 7f 32 d6 68 = z8

w32 = w28 ⊕ z8 = 48 26 45 20
w33 = w32 ⊕ w29 = f3 1b a2 d7
w34 = w33 ⊕ w30 = cb c3 aa 72
w35 = w34 ⊕ w32 = 3c be 0b 3

RotWord (w35) = be 0b 38 3c = x9
SubWord (x9) = ae 2b 07 eb = y9
Rcon (9) = 1B 00 00 00
y9 ⊕ Rcon (9) = b5 2b 07 eb = z9

w36 = w32 ⊕ z9 = fd 0d 42 cb
w37 = w36 ⊕ w33 = 0e 16 e0 1c
w38 = w37 ⊕ w34 = c5 d5 4a 6e
w39 = w38 ⊕ w35 = f9 6b 41 56

RotWord (w39) = 6b 41 56 f9 = x10
SubWord (x10) = 7f 83 b1 99 = y10
Rcon (10) = 36 00 00 00
y10 ⊕ Rcon (10) = 49 83 b1 99 = z10

w40 = w36 ⊕ z10 = b4 8e f3 52
w41 = w40 ⊕ w37 = ba 98 13 4e
w42 = w41 ⊕ w38 = 7f 4d 59 20
w43 = w42 ⊕ w39 = 86 26 18 76

Table 6.3 Continued

the four round-key words generated for each round. The right-hand column shows
the steps used to generate the auxiliary word used in key expansion. We begin, of
course, with the key itself serving as the round key for round 0.

Next, Table 6.4 shows the progression of State through the AES encryption
process. The first column shows the value of State at the start of a round. For the
first row, State is just the matrix arrangement of the plaintext. The second, third, and
fourth columns show the value of State for that round after the SubBytes, ShiftRows,
and MixColumns transformations, respectively. The fifth column shows the round
key. You can verify that these round keys equate with those shown in Table 6.3. The
first column shows the value of State resulting from the bitwise XOR of State after
the preceding MixColumns with the round key for the preceding round.

Avalanche Effect

If a small change in the key or plaintext were to produce a corresponding small
change in the ciphertext, this might be used to effectively reduce the size of the

6.5 / AN AES EXAMPLE 195

Start of Round After SubBytes After ShiftRows After MixColumns Round Key

01 89 fe 76
23 ab dc 54
45 cd ba 32
67 ef 98 10

0f 47 0c af
15 d9 b7 7f
71 e8 ad 67
c9 59 d6 98

0e ce f2 d9
36 72 6b 2b
34 25 17 55
ae b6 4e 88

ab 8b 89 35
05 40 7f f1
18 3f f0 fc
e4 4e 2f c4

ab 8b 89 35
40 7f f1 05
f0 fc 18 3f
c4 e4 4e 2f

b9 94 57 75
e4 8e 16 51
47 20 9a 3f
c5 d6 f5 3b

dc 9b 97 38
90 49 fe 81
37 df 72 15
b0 e9 3f a7

65 0f c0 4d
74 c7 e8 d0
70 ff e8 2a
75 3f ca 9c

4d 76 ba e3
92 c6 9b 70
51 16 9b e5
9d 75 74 de

4d 76 ba e3
c6 9b 70 92
9b e5 51 16
de 9d 75 74

8e 22 db 12
b2 f2 dc 92
df 80 f7 c1
2d c5 1e 52

d2 49 de e6
c9 80 7e ff
6b b4 c6 d3
b7 5e 61 c6

5c 6b 05 f4
7b 72 a2 6d
b4 34 31 12
9a 9b 7f 94

4a 7f 6b bf
21 40 3a 3c
8d 18 c7 c9
b8 14 d2 22

4a 7f 6b bf
40 3a 3c 21
c7 c9 8d 18
22 b8 14 d2

b1 c1 0b cc
ba f3 8b 07
f9 1f 6a c3
1d 19 24 5c

c0 89 57 b1
af 2f 51 ae
df 6b ad 7e
39 67 06 c0

71 48 5c 7d
15 dc da a9
26 74 c7 bd
24 7e 22 9c

a3 52 4a ff
59 86 57 d3
f7 92 c6 7a
36 f3 93 de

a3 52 4a ff
86 57 d3 59
c6 7a f7 92
de 36 f3 93

d4 11 fe 0f
3b 44 06 73
cb ab 62 37
19 b7 07 ec

2c a5 f2 43
5c 73 22 8c
65 0e a3 dd
f1 96 90 50

f8 b4 0c 4c
67 37 24 ff
ae a5 c1 ea
e8 21 97 bc

41 8d fe 29
85 9a 36 16
e4 06 78 87
9b fd 88 65

41 8d fe 29
9a 36 16 85
78 87 e4 06
65 9b fd 88

2a 47 c4 48
83 e8 18 ba
84 18 27 23
eb 10 0a f3

58 fd 0f 4c
9d ee cc 40
36 38 9b 46
eb 7d ed bd

72 ba cb 04
1e 06 d4 fa
b2 20 bc 65
00 6d e7 4e

40 f4 1f f2
72 6f 48 2d
37 b7 65 4d
63 3c 94 2f

40 f4 1f f2
6f 48 2d 72
65 4d 37 b7
2f 63 3c 94

7b 05 42 4a
1e d0 20 40
94 83 18 52
94 c4 43 fb

71 8c 83 cf
c7 29 e5 a5
4c 74 ef a9
c2 bf 52 ef

0a 89 c1 85
d9 f9 c5 e5
d8 f7 f7 fb
56 7b 11 14

67 a7 78 97
35 99 a6 d9
61 68 68 0f
b1 21 82 fa

67 a7 78 97
99 a6 d9 35
68 0f 61 68
fa b1 21 82

ec 1a c0 80
0c 50 53 c7
3b d7 00 ef
b7 22 72 e0

37 bb 38 f7
14 3d d8 7d
93 e7 08 a1
48 f7 a5 4a

db a1 f8 77
18 6d 8b ba
a8 30 08 4e
ff d5 d7 aa

b9 32 41 f5
ad 3c 3d f4
c2 04 30 2f
16 03 0e ac

b9 32 41 f5
3c 3d f4 ad
30 2f c2 04
ac 16 03 0e

b1 1a 44 17
3d 2f ec b6
0a 6b 2f 42
9f 68 f3 b1

48 f3 cb 3c
26 1b c3 be
45 a2 aa 0b
20 d7 72 38

f9 e9 8f 2b
1b 34 2f 08
4f c9 85 49
bf bf 81 89

99 1e 73 f1
af 18 15 30
84 dd 97 3b
08 08 0c a7

99 1e 73 f1
18 15 30 af
97 3b 84 dd
a7 08 08 0c

31 30 3a c2
ac 71 8c c4
46 65 48 eb
6a 1c 31 62

fd 0e c5 f9
0d 16 d5 6b
42 e0 4a 41
cb 1c 6e 56

cc 3e ff 3b
a1 67 59 af
04 85 02 aa
a1 00 5f 34

4b b2 16 e2
32 85 cb 79
f2 97 77 ac
32 63 cf 18

4b b2 16 e2
85 cb 79 32
77 ac f2 97
18 32 63 cf

b4 ba 7f 86
8e 98 4d 26
f3 13 59 18
52 4e 20 76

ff 08 69 64
0b 53 34 14
84 bf ab 8f
4a 7c 43 b9

Table 6.4 AES Example

196 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Round
Number of Bits

that Differ

0123456789abcdeffedcba9876543210
0023456789abcdeffedcba9876543210

1

0 0e3634aece7225b6f26b174ed92b5588
0f3634aece7225b6f26b174ed92b5588

1

1 657470750fc7ff3fc0e8e8ca4dd02a9c
c4a9ad090fc7ff3fc0e8e8ca4dd02a9c

20

2 5c7bb49a6b72349b05a2317ff46d1294
fe2ae569f7ee8bb8c1f5a2bb37ef53d5

58

3 7115262448dc747e5cdac7227da9bd9c
ec093dfb7c45343d689017507d485e62

59

4 f867aee8b437a5210c24c1974cffeabc
43efdb697244df808e8d9364ee0ae6f5

61

5 721eb200ba06206dcbd4bce704fa654e
7b28a5d5ed643287e006c099bb375302

68

6 0ad9d85689f9f77bc1c5f71185e5fb14
3bc2d8b6798d8ac4fe36a1d891ac181a

64

7 db18a8ffa16d30d5f88b08d777ba4eaa
9fb8b5452023c70280e5c4bb9e555a4b

67

8 f91b4fbfe934c9bf8f2f85812b084989
20264e1126b219aef7feb3f9b2d6de40

65

9 cca104a13e678500ff59025f3bafaa34
b56a0341b2290ba7dfdfbddcd8578205

61

10 ff0b844a0853bf7c6934ab4364148fb9
612b89398d0600cde116227ce72433f0

58

Table 6.5 Avalanche Effect in AES: Change in Plaintext

plaintext (or key) space to be searched. What is desired is the avalanche effect, in
which a small change in plaintext or key produces a large change in the ciphertext.

Using the example from Table 6.4, Table 6.5 shows the result when the
eighth bit of the plaintext is changed. The second column of the table shows the
value of the State matrix at the end of each round for the two plaintexts. Note
that after just one round, 20 bits of the State vector differ. After two rounds,
close to half the bits differ. This magnitude of difference propagates through
the remaining rounds. A bit difference in approximately half the positions in the
most desirable outcome. Clearly, if almost all the bits are changed, this would be
logically equivalent to almost none of the bits being changed. Put another way, if
we select two plaintexts at random, we would expect the two plaintexts to differ
in about half of the bit positions and the two ciphertexts to also differ in about
half the positions.

Table 6.6 shows the change in State matrix values when the same plaintext
is used and the two keys differ in the eighth bit. That is, for the second case, the
key is 0e1571c947d9e8590cb7add6af7f6798. Again, one round produces
a significant change, and the magnitude of change after all subsequent rounds
is roughly half the bits. Thus, based on this example, AES exhibits a very strong
avalanche effect.

6.6 / AES IMPLEMENTATION 197

Round
Number of Bits

that Differ

0123456789abcdeffedcba9876543210
0123456789abcdeffedcba9876543210

0

0 0e3634aece7225b6f26b174ed92b5588
0f3634aece7225b6f26b174ed92b5588

1

1 657470750fc7ff3fc0e8e8ca4dd02a9c
c5a9ad090ec7ff3fc1e8e8ca4cd02a9c

22

2 5c7bb49a6b72349b05a2317ff46d1294
90905fa9563356d15f3760f3b8259985

58

3 7115262448dc747e5cdac7227da9bd9c
18aeb7aa794b3b66629448d575c7cebf

67

4 f867aee8b437a5210c24c1974cffeabc
f81015f993c978a876ae017cb49e7eec

63

5 721eb200ba06206dcbd4bce704fa654e
5955c91b4e769f3cb4a94768e98d5267

81

6 0ad9d85689f9f77bc1c5f71185e5fb14
dc60a24d137662181e45b8d3726b2920

70

7 db18a8ffa16d30d5f88b08d777ba4eaa
fe8343b8f88bef66cab7e977d005a03c

74

8 f91b4fbfe934c9bf8f2f85812b084989
da7dad581d1725c5b72fa0f9d9d1366a

67

9 cca104a13e678500ff59025f3bafaa34
0ccb4c66bbfd912f4b511d72996345e0

59

10 ff0b844a0853bf7c6934ab4364148fb9
fc8923ee501a7d207ab670686839996b

53

Table 6.6 Avalanche Effect in AES: Change in Key

Note that this avalanche effect is stronger than that for DES (Table 4.2),
which requires three rounds to reach a point at which approximately half the bits
are changed, both for a bit change in the plaintext and a bit change in the key.

6.6 AES IMPLEMENTATION

Equivalent Inverse Cipher

As was mentioned, the AES decryption cipher is not identical to the encryption
cipher (Figure 6.3). That is, the sequence of transformations for decryption differs
from that for encryption, although the form of the key schedules for encryption
and decryption is the same. This has the disadvantage that two separate software
or firmware modules are needed for applications that require both encryption and
decryption. There is, however, an equivalent version of the decryption algorithm
that has the same structure as the encryption algorithm. The equivalent version has
the same sequence of transformations as the encryption algorithm (with transfor-
mations replaced by their inverses). To achieve this equivalence, a change in key
schedule is needed.

198 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

Two separate changes are needed to bring the decryption structure in line
with the encryption structure. As illustrated in Figure 6.3, an encryption round has
the structure SubBytes, ShiftRows, MixColumns, AddRoundKey. The standard
decryption round has the structure InvShiftRows, InvSubBytes, AddRoundKey,
InvMixColumns. Thus, the first two stages of the decryption round need to be inter-
changed, and the second two stages of the decryption round need to be interchanged.

INTERCHANGING INVSHIFTROWS AND INVSUBBYTES InvShiftRows affects the se-
quence of bytes in State but does not alter byte contents and does not depend on
byte contents to perform its transformation. InvSubBytes affects the contents of
bytes in State but does not alter byte sequence and does not depend on byte se-
quence to perform its transformation. Thus, these two operations commute and can
be interchanged. For a given State Si,

InvShiftRows [InvSubBytes (Si)] = InvSubBytes [InvShiftRows (Si)]

INTERCHANGING ADDROUNDKEY AND INVMIXCOLUMNS The transformations
AddRoundKey and InvMixColumns do not alter the sequence of bytes in State. If we
view the key as a sequence of words, then both AddRoundKey and InvMixColumns
operate on State one column at a time. These two operations are linear with respect
to the column input. That is, for a given State Si and a given round key wj,

InvMixColumns (Si⊕ wj) = [InvMixColumns (Si)]⊕ [InvMixColumns (wj)]

To see this, suppose that the first column of State Si is the sequence (y0, y1, y2, y3)
and the first column of the round key wj is (k0, k1, k2, k3). Then we need to show

D 0E 0B 0D 0909 0E 0B 0D
0D 09 0E 0B
0B 0D 09 0E

T Dy0⊕ k0y1⊕ k1
y2⊕ k2
y3⊕ k3

T = D 0E 0B 0D 0909 0E 0B 0D
0D 09 0E 0B
0B 0D 09 0E

T Dy0y1
y2
y3

T ⊕ D 0E 0B 0D 0909 0E 0B 0D
0D 09 0E 0B
0B 0D 09 0E

T Dk0k1
k2
k3

T
Let us demonstrate that for the first column entry. We need to show

[{0E} # (y0⊕ k0)]⊕ [{0B} # (y1⊕ k1)]⊕ [{0D} # (y2⊕ k2)]⊕ [{09} # (y3⊕ k3)]
= [{0E} # y0]⊕ [{0B} # y1]⊕ [{0D} # y2]⊕ [{09} # y3]⊕

[{0E} # k0]⊕ [{0B} # k1]⊕ [{0D} # k2]⊕ [{09} # k3]
This equation is valid by inspection. Thus, we can interchange AddRoundKey

and InvMixColumns, provided that we first apply InvMixColumns to the round
key. Note that we do not need to apply InvMixColumns to the round key for the
input to the first AddRoundKey transformation (preceding the first round) nor to
the last AddRoundKey transformation (in round 10). This is because these two
AddRoundKey transformations are not interchanged with InvMixColumns to pro-
duce the equivalent decryption algorithm.

Figure 6.10 illustrates the equivalent decryption algorithm.

6.6 / AES IMPLEMENTATION 199

Figure 6.10 Equivalent Inverse Cipher

Add round key

w[36, 39]

w[40, 43]

Ciphertext

Inverse sub bytes

Inverse shift rows

Inverse mix cols R
ou

nd
1

R
ou

nd
9

R
ou

nd
1

0

Add round keyInverse mix cols

Inverse sub bytes

Inverse shift rows

Inverse mix cols

Add round keyInverse mix cols

Inverse sub bytes

Inverse shift rowsExpand key

Add round key

PlaintextKey

w[4, 7]

w[0, 3]

Implementation Aspects

The Rijndael proposal [DAEM99] provides some suggestions for efficient im-
plementation on 8-bit processors, typical for current smart cards, and on 32-bit
processors, typical for PCs.

8-BIT PROCESSOR AES can be implemented very efficiently on an 8-bit proces-
sor. AddRoundKey is a bytewise XOR operation. ShiftRows is a simple byte-
shifting operation. SubBytes operates at the byte level and only requires a table
of 256 bytes.

The transformation MixColumns requires matrix multiplication in the field
GF(28), which means that all operations are carried out on bytes. MixColumns only
requires multiplication by {02} and {03}, which, as we have seen, involved simple
shifts, conditional XORs, and XORs. This can be implemented in a more efficient

Hiva-Network.Com

200 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

way that eliminates the shifts and conditional XORs. Equation set (6.4) shows the
equations for the MixColumns transformation on a single column. Using the iden-
tity {03} # x = ({02} # x)⊕ x, we can rewrite Equation set (6.4) as follows.

Tmp = s0, j⊕ s1, j⊕ s2, j⊕ s3, j
s0, j
= = s0, j⊕ Tmp⊕ [2 # (s0, j⊕ s1, j)]

s1, j
= = s1, j⊕ Tmp⊕ [2 # (s1, j⊕ s2, j)] (6.9)

s2, j
= = s2, j⊕ Tmp⊕ [2 # (s2, j⊕ s3, j)]

s3, j
= = s3, j⊕ Tmp⊕ [2 # (s3, j⊕ s0, j)]

Equation set (6.9) is verified by expanding and eliminating terms.
The multiplication by {02} involves a shift and a conditional XOR. Such

an implementation may be vulnerable to a timing attack of the sort described in
Section 4.4. To counter this attack and to increase processing efficiency at the
cost of some storage, the multiplication can be replaced by a table lookup. Define
the 256-byte table X2, such that X2[i] = {02} # i. Then Equation set (6.9) can be
rewritten as

Tmp = s0, j⊕ s1, j⊕ s2, j⊕ s3, j
s0, j
= = s0, j⊕ Tmp⊕ X2[s0, j⊕ s1, j]

s1, c
= = s1, j⊕ Tmp⊕ X2[s1, j⊕ s2, j]

s2, c
= = s2, j⊕ Tmp⊕ X2[s2, j⊕ s3, j]

s3, j
= = s3, j⊕ Tmp⊕ X2[s3, j⊕ s0, j]

32-BIT PROCESSOR The implementation described in the preceding subsection uses
only 8-bit operations. For a 32-bit processor, a more efficient implementation can be
achieved if operations are defined on 32-bit words. To show this, we first define the
four transformations of a round in algebraic form. Suppose we begin with a State
matrix consisting of elements ai, j and a round-key matrix consisting of elements ki, j.
Then the transformations can be expressed as follows.

SubBytes bi, j = S[ai, j]

ShiftRows D c0, jc1, j
c2, j
c3, j

T = D b0, jb1, j- 1
b2, j- 2
b3, j- 3

T
MixColumns Dd0, jd1, j

d2, j
d3, j

T = D02 03 01 0101 02 03 01
01 01 02 03
03 01 01 02

T D c0, jc1, j
c2, j
c3, j

T
AddRoundKey D e0, je1, j

e2, j
e3, j

T = Dd0, jd1, j
d2, j
d3, j

T ⊕ Dk0, jk1, j
k2, j
k3, j

T

6.6 / AES IMPLEMENTATION 201

In the ShiftRows equation, the column indices are taken mod 4. We can
combine all of these expressions into a single equation:

D e0, je1, j
e2, j
e3, j

T = D02 03 01 0101 02 03 01
01 01 02 03
03 01 01 02

T D S[a0, j]S[a1, j- 1]
S[a2, j- 2]
S[a3, j- 3]

T ⊕ Dk0, jk1, j
k2, j
k3, j

T
= § D0201

01
03

T # S[a0, j]¥ ⊕ § D030201
01

T # S[a1, j- 1]¥ ⊕ § D010302
01

T # S[a2, j- 2]¥
⊕ § D0101

03
02

T # S[a3, j- 3]¥ ⊕ Dk0, jk1, jk2, j
k3, j

T
In the second equation, we are expressing the matrix multiplication as a linear com-
bination of vectors. We define four 256-word (1024-byte) tables as follows.

T0[x] = § D020101
03

T # S[x]¥ T1[x] = § D030201
01

T # S[x]¥ T2[x] = § D010302
01

T # S[x]¥ T3[x] = § D010103
02

T # S[x]¥
Thus, each table takes as input a byte value and produces a column vector (a 32-bit
word) that is a function of the S-box entry for that byte value. These tables can be
calculated in advance.

We can define a round function operating on a column in the following fashion.

D s0, j=s1, j=
s2, j
=

s3, j
=

T = T0[s0, j]⊕ T1[s1, j- 1]⊕ T2[s2, j- 2]⊕ T3[s3, j- 3]⊕ Dk0, jk1, jk2, j
k3, j

T
As a result, an implementation based on the preceding equation requires only

four table lookups and four XORs per column per round, plus 4 Kbytes to store the
table. The developers of Rijndael believe that this compact, efficient implementa-
tion was probably one of the most important factors in the selection of Rijndael
for AES.

202 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

6.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Advanced Encryption
Standard (AES)

avalanche effect
field

finite field
irreducible

polynomial
key expansion

National Institute of Standards
and Technology (NIST)

Rijndael
S-box

Key Terms

Review Questions

6.1 What was the original set of criteria used by NIST to evaluate candidate AES ciphers?
6.2 What was the final set of criteria used by NIST to evaluate candidate AES ciphers?
6.3 What is the difference between Rijndael and AES?
6.4 What is the purpose of the State array?
6.5 How is the S-box constructed?
6.6 Briefly describe SubBytes.
6.7 Briefly describe ShiftRows.
6.8 How many bytes in State are affected by ShiftRows?
6.9 Briefly describe MixColumns.
6.10 Briefly describe AddRoundKey.
6.11 Briefly describe the key expansion algorithm.
6.12 What is the difference between SubBytes and SubWord?
6.13 What is the difference between ShiftRows and RotWord?
6.14 What is the difference between the AES decryption algorithm and the equivalent

inverse cipher?

Problems

6.1 In the discussion of MixColumns and InvMixColumns, it was stated that

b(x) = a-1(x) mod(x4 + 1)

where a(x) = {03}x3 + {01}x2 + {01}x + {02} and b(x) = {0B}x3 + {0D}x2 + {09}x +
{0E.} Show that this is true.

6.2 a. What is {0 2 }-1 in GF(28)?
b. Verify the entry for {0 2 } in the S-box.

6.3 Show the first eight words of the key expansion for a 128-bit key of all ones.
6.4 Given the plaintext {0F0E0D0C0B0A09080706050403020100} and the key

{02020202020202020202020202020202}:
a. Show the original contents of State, displayed as a 4 * 4 matrix.
b. Show the value of State after initial AddRoundKey.
c. Show the value of State after SubBytes.
d. Show the value of State after ShiftRows.
e. Show the value of State after MixColumns.

6.5 Verify Equation (6.11) in Appendix 6A. That is, show that xi mod (x4 + 1) = xi mod 4.

APPENDIX 6A / POLYNOMIALS WITH COEFFICIENTS IN GF(28) 203

6.6 Compare AES to DES. For each of the following elements of DES, indicate the com-
parable element in AES or explain why it is not needed in AES.
a. XOR of subkey material with the input to the f function
b. XOR of the f function output with the left half of the block
c. f function
d. permutation P
e. swapping of halves of the block

6.7 In the subsection on implementation aspects, it is mentioned that the use of tables
helps thwart timing attacks. Suggest an alternative technique.

6.8 In the subsection on implementation aspects, a single algebraic equation is developed
that describes the four stages of a typical round of the encryption algorithm. Provide
the equivalent equation for the tenth round.

6.9 Compute the output of the MixColumns transformation for the following sequence
of input bytes “A1 B2 C3 D4.” Apply the InvMixColumns transformation to the ob-
tained result to verify your calculations. Change the first byte of the input from “A1”
to “A3” perform the MixColumns transformation again for the new input, and deter-
mine how many bits have changed in the output.

Note: You can perform all calculations by hand or write a program supporting these
computations. If you choose to write a program, it should be written entirely by you;
no use of libraries or public domain source code is allowed in this assignment.

6.10 Use the key 1010 1001 1100 0011 to encrypt the plaintext “hi” as expressed in ASCII
as 0110 1000 0110 1001. The designers of S-AES got the ciphertext 0011 1110 1111
1011. Do you?

6.11 Show that the matrix given here, with entries in GF(24), is the inverse of the matrix
used in the MixColumns step of S-AES.¢x3 + 1 x

x x3 + 1
≤

6.12 Carefully write up a complete decryption of the ciphertext 0011 1110 1111 1011 using
the key 1010 1001 1100 0011 and the S-AES algorithm. You should get the plaintext
we started with in Problem 6.10. Note that the inverse of the S-boxes can be done
with a reverse table lookup. The inverse of the MixColumns step is given by the ma-
trix in the previous problem.

6.13 Demonstrate that Equation (6.9) is equivalent to Equation (6.4).

Programming Problems

6.14 Create software that can encrypt and decrypt using S-AES. Test data: A binary
plaintext of 0110 1111 0110 1011 encrypted with a binary key of 1010 0111 0011 1011
should give a binary ciphertext of 0000 0111 0011 1000. Decryption should work
correspondingly.

6.15 Implement a differential cryptanalysis attack on 1-round S-AES.

APPENDIX 6A POLYNOMIALS WITH COEFFICIENTS IN GF(28)

In Section 5.5, we discussed polynomial arithmetic in which the coefficients are in Zp
and the polynomials are defined modulo a polynomial m(x) whose highest power
is some integer n. In this case, addition and multiplication of coefficients occurred
within the field Zp; that is, addition and multiplication were performed modulo p.

204 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

The AES document defines polynomial arithmetic for polynomials of degree 3
or less with coefficients in GF(28). The following rules apply.

1. Addition is performed by adding corresponding coefficients in GF(28). As was
pointed out Section 5.4, if we treat the elements of GF(28) as 8-bit strings, then
addition is equivalent to the XOR operation. So, if we have

a(x) = a3x3 + a2x2 + a1x + a0 (6.10)

and

b(x) = b3x3 + b2x2 + b1x + b0 (6.11)

then

a(x) + b(x) = (a3⊕ b3)x3 + (a2⊕ b2)x2 + (a1 ⊕ b1)x + (a0⊕ b0)

2. Multiplication is performed as in ordinary polynomial multiplication with two
refinements:

a. Coefficients are multiplied in GF(28).
b. The resulting polynomial is reduced mod (x4 + 1).

We need to keep straight which polynomial we are talking about. Recall from
Section 5.6 that each element of GF(28) is a polynomial of degree 7 or less with bi-
nary coefficients, and multiplication is carried out modulo a polynomial of degree
8. Equivalently, each element of GF(28) can be viewed as an 8-bit byte whose bit
values correspond to the binary coefficients of the corresponding polynomial. For
the sets defined in this section, we are defining a polynomial ring in which each ele-
ment of this ring is a polynomial of degree 3 or less with coefficients in GF(28), and
multiplication is carried out modulo a polynomial of degree 4. Equivalently, each
element of this ring can be viewed as a 4-byte word whose byte values are elements
of GF(28) that correspond to the 8-bit coefficients of the corresponding polynomial.

We denote the modular product of a(x) and b(x) by a(x)⊕ b(x). To com-
pute d(x) = a(x)⊕ b(x), the first step is to perform a multiplication without the
modulo operation and to collect coefficients of like powers. Let us express this as
c(x) = a(x) * b(x). Then

c(x) = c6x6 + c5x5 + c4x4 + c3x3 + c2x2 + c1x + c0 (6.12)

where

c0 = a0 # b0 c4 = (a3 # b1)⊕ (a2 # b2)⊕ (a1 # b3)
c1 = (a1 # b0)⊕ (a0 # b1) c5 = (a3 # b2)⊕ (a2 # b3)
c2 = (a2 # b0)⊕ (a1 # b1)⊕ (a0 # b2) c6 = a3 # b3
c3 = (a3 # b0)⊕ (a2 # b1)⊕ (a1 # b2)⊕ (a0 # b3)

The final step is to perform the modulo operation

d(x) = c(x) mod (x4 + 1)

That is, d(x) must satisfy the equation

c(x) = [(x4 + 1) * q(x)]⊕ d(x)

such that the degree of d(x) is 3 or less.
A practical technique for performing multiplication over this polynomial ring

is based on the observation that

xi mod (x4 + 1) = xi mod 4 (6.13)

If we now combine Equations (6.12) and (6.13), we end up with

d(x) = c(x) mod (x4 + 1)
= [c6x6 + c5x5 + c4x4 + c3x3 + c2x2 + c1x + c0] mod (x4 + 1)
= c3x3 + (c2⊕ c6)x2 + (c1⊕ c5)x + (c0⊕ c4)

Expanding the ci coefficients, we have the following equations for the coef-
ficients of d(x).

d0 = (a0 # b0)⊕ (a3 # b1)⊕ (a2 # b2)⊕ (a1 # b3)
d1 = (a1 # b0)⊕ (a0 # b1)⊕ (a3 # b2)⊕ (a2 # b3)
d2 = (a2 # b0)⊕ (a1 # b1)⊕ (a0 # b2)⊕ (a3 # b3)
d3 = (a3 # b0)⊕ (a2 # b1)⊕ (a1 # b2)⊕ (a0 # b3)

This can be written in matrix form:

Dd0d1
d2
d3

T = Da0 a3 a2 a1a1 a0 a3 a2
a2 a1 a0 a3
a3 a2 a1 a0

T Db0b1
b2
b3

T (6.14)
MixColumns Transformation

In the discussion of MixColumns, it was stated that there were two equivalent
ways of defining the transformation. The first is the matrix multiplication shown in
Equation (6.3), which is repeated here:

D02 03 01 0101 02 03 01
01 01 02 03
03 01 01 02

T D s0, 0 s0, 1 s0, 2 s0, 3s1, 0 s1, 1 s1, 2 s1, 3
s2, 0 s2, 1 s2, 2 s2, 3
s3, 0 s3, 1 s3, 2 s3, 3

T = D s0, 0= s0, 1= s0, 2= s0, 3=s1, 0= s1, 1= s1, 2= s1, 3=
s2, 0
= s2, 1

= s2, 2
= s2, 3

=

s3, 0
= s3, 1

= s3, 2
= s3, 3

=

T
The second method is to treat each column of State as a four-term polynomial

with coefficients in GF(28). Each column is multiplied modulo (x4 + 1) by the fixed
polynomial a(x), given by

a(x) = {03}x3 + {01}x2 + {01}x + {02}

APPENDIX 6A / POLYNOMIALS WITH COEFFICIENTS IN GF(28) 205

206 CHAPTER 6 / ADVANCED ENCRYPTION STANDARD

From Equation (6.10), we have a3 = {03}; a2 = {01}; a1 = {01}; and
a0 = {02}. For the jth column of State, we have the polynomial colj(x) = s3,jx3 +
s2,jx

2 + s1,jx + s0, j. Substituting into Equation (6.14), we can express
d(x) = a(x) * colj(x) as

Dd0d1
d2
d3

T = Da0 a3 a2 a1a1 a0 a3 a2
a2 a1 a0 a3
a3 a2 a1 a0

T D s0,js1,j
s2,j
s3,j

T = D02 03 01 0101 02 03 01
01 01 02 03
03 01 01 02

T D s0,js1,j
s2,j
s3,j

T
which is equivalent to Equation (6.3).

Multiplication by x

Consider the multiplication of a polynomial in the ring by x: c(x) = x⊕ b(x).
We have

c(x) = x⊕ b(x) = [x * (b3x3 + b2x2 + b1x + b0] mod (x4 + 1)
= (b3x4 + b2x3 + b1x2 + b0x) mod (x4 + 1)
= b2x3 + b1x2 + b0x + b3

Thus, multiplication by x corresponds to a 1-byte circular left shift of the
4 bytes in the word representing the polynomial. If we represent the polynomial as
a 4-byte column vector, then we have

D c0c1
c2
c3

T = D00 00 00 0101 00 00 00
00 01 00 00
00 00 01 00

T Db0b1
b2
b3

T

207

Block Cipher Operation
7.1 Multiple Encryption and Triple DES

Double DES
Triple DES with Two Keys
Triple DES with Three Keys

7.2 Electronic Codebook

7.3 Cipher Block Chaining Mode

7.4 Cipher Feedback Mode

7.5 Output Feedback Mode

7.6 Counter Mode

7.7 XTS-AES Mode for Block-Oriented Storage Devices

Tweakable Block Ciphers
Storage Encryption Requirements
Operation on a Single Block
Operation on a Sector

7.8 Format-Preserving Encryption

Motivation
Difficulties in Designing an FPE
Feistel Structure for Format-Preserving Encryption
NIST Methods for Format-Preserving Encryption

7.9 Key Terms, Review Questions, and Problems

CHAPTER

208 CHAPTER 7 / BLOCK CIPHER OPERATION

This chapter continues our discussion of symmetric ciphers. We begin with the topic of
multiple encryption, looking in particular at the most widely used multiple-encryption
scheme: triple DES.

The chapter next turns to the subject of block cipher modes of operation. We
find that there are a number of different ways to apply a block cipher to plaintext, each
with its own advantages and particular applications.

7.1 MULTIPLE ENCRYPTION AND TRIPLE DES

Because of its vulnerability to brute-force attack, DES, once the most widely used
symmetric cipher, has been largely replaced by stronger encryption schemes. Two
approaches have been taken. One approach is to design a completely new algo-
rithm that is resistant to both cryptanalytic and brute-force attacks, of which AES
is a prime example. Another alternative, which preserves the existing investment in
software and equipment, is to use multiple encryption with DES and multiple keys.
We begin by examining the simplest example of this second alternative. We then
look at the widely accepted triple DES (3DES) algorithm.

Double DES

The simplest form of multiple encryption has two encryption stages and two keys
(Figure 7.1a). Given a plaintext P and two encryption keys K1 and K2, ciphertext C
is generated as

C = E(K2, E(K1, P))

Decryption requires that the keys be applied in reverse order:

P = D(K1, D(K2, C))

For DES, this scheme apparently involves a key length of 56 * 2 = 112 bits, and
should result in a dramatic increase in cryptographic strength. But we need to exam-
ine the algorithm more closely.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Analyze the security of multiple encryption schemes.

◆ Explain the meet-in-the-middle attack.

◆ Compare and contrast ECB, CBC, CFB, OFB, and counter modes of operation.

◆ Present an overview of the XTS-AES mode of operation.

Hiva-Network.Com

7.1 / MULTIPLE ENCRYPTION AND TRIPLE DES 209

REDUCTION TO A SINGLE STAGE Suppose it were true for DES, for all 56-bit key val-
ues, that given any two keys K1 and K2, it would be possible to find a key K3 such that

E(K2, E(K1, P)) = E(K3, P) (7.1)

If this were the case, then double encryption, and indeed any number of stages of
multiple encryption with DES, would be useless because the result would be equiv-
alent to a single encryption with a single 56-bit key.

On the face of it, it does not appear that Equation (7.1) is likely to hold.
Consider that encryption with DES is a mapping of 64-bit blocks to 64-bit blocks.
In fact, the mapping can be viewed as a permutation. That is, if we consider all 264
possible input blocks, DES encryption with a specific key will map each block into a
unique 64-bit block. Otherwise, if, say, two given input blocks mapped to the same
output block, then decryption to recover the original plaintext would be impossible.

Figure 7.1 Multiple Encryption

(3-key)

(2-key)K1

K3
or

(3-key)

(2-key)K1

K3
or

E E

K1

P

K2

C
X

Encryption

D D

K1

C

K2

P
X

Decryption
(a) Double encryption

E D E

K1

P

K2

C
A B

Encryption

D E D

K1

C

K2

P

Decryption
(b) Triple encryption

B A

210 CHAPTER 7 / BLOCK CIPHER OPERATION

With 264 possible inputs, how many different mappings are there that generate a
permutation of the input blocks? The value is easily seen to be

(264)! = 10347380000000000000000 7 (1010
20
)

On the other hand, DES defines one mapping for each different key, for a total
number of mappings:

256 6 1017

Therefore, it is reasonable to assume that if DES is used twice with different keys, it
will produce one of the many mappings that are not defined by a single application
of DES. Although there was much supporting evidence for this assumption, it was
not until 1992 that the assumption was proven [CAMP92].

MEET-IN-THE-MIDDLE ATTACK Thus, the use of double DES results in a mapping
that is not equivalent to a single DES encryption. But there is a way to attack this
scheme, one that does not depend on any particular property of DES but that will
work against any block encryption cipher.

The algorithm, known as a meet-in-the-middle attack, was first described in
[DIFF77]. It is based on the observation that, if we have

C = E(K2, E(K1, P))

then (see Figure 7.1a)

X = E(K1, P) = D(K2, C)

Given a known pair, (P, C), the attack proceeds as follows. First, encrypt P for all
256 possible values of K1. Store these results in a table and then sort the table by the
values of X. Next, decrypt C using all 256 possible values of K2. As each decryption
is produced, check the result against the table for a match. If a match occurs, then
test the two resulting keys against a new known plaintext–ciphertext pair. If the two
keys produce the correct ciphertext, accept them as the correct keys.

For any given plaintext P, there are 264 possible ciphertext values that could be
produced by double DES. Double DES uses, in effect, a 112-bit key, so that there
are 2112 possible keys. Therefore, for a given plaintext P, the maximum number
of different 112-bit keys that could produce a given ciphertext C is 2112/264 = 248.
Thus, the foregoing procedure can produce about 248 false alarms on the first (P, C)
pair. A similar argument indicates that with an additional 64 bits of known plaintext
and ciphertext, the false alarm rate is reduced to 248 - 64 = 2-16. Put another way,
if the meet-in-the-middle attack is performed on two blocks of known plaintext–
ciphertext, the probability that the correct keys are determined is 1 - 2-16. The
result is that a known plaintext attack will succeed against double DES, which has a
key size of 112 bits, with an effort on the order of 256, which is not much more than
the 255 required for single DES.

Triple DES with Two Keys

An obvious counter to the meet-in-the-middle attack is to use three stages of
encryption with three different keys. Using DES as the underlying algorithm,
this approach is commonly referred to as 3DES, or Triple Data Encryption

7.1 / MULTIPLE ENCRYPTION AND TRIPLE DES 211

Algorithm (TDEA). As shown in Figure 7.1b, there are two versions of 3DES;
one using two keys and one using three keys. NIST SP 800-67 (Recommendation
for the Triple Data Encryption Block Cipher, January 2012) defines the two-key
and three-key versions. We look first at the strength of the two-key version and
then examine the three-key version.

Two-key triple encryption was first proposed by Tuchman [TUCH79]. The
function follows an encrypt-decrypt-encrypt (EDE) sequence (Figure 7.1b):

C = E(K1, D(K2, E(K1, P)))
P = D(K1, E(K2, D(K1, C)))

There is no cryptographic significance to the use of decryption for the second
stage. Its only advantage is that it allows users of 3DES to decrypt data encrypted by
users of the older single DES:

C = E(K1, D(K1, E(K1, P))) = E(K1, P)
P = D(K1, E(K1, D(K1, C))) = D(K1, C)

3DES with two keys is a relatively popular alternative to DES and has been
adopted for use in the key management standards ANSI X9.17 and ISO 8732.1

Currently, there are no practical cryptanalytic attacks on 3DES. Coppersmith
[COPP94] notes that the cost of a brute-force key search on 3DES is on the order of
2112 ≈ (5 * 1033) and estimates that the cost of differential cryptanalysis suffers an
exponential growth, compared to single DES, exceeding 1052.

It is worth looking at several proposed attacks on 3DES that, although not
practical, give a flavor for the types of attacks that have been considered and that
could form the basis for more successful future attacks.

The first serious proposal came from Merkle and Hellman [MERK81]. Their
plan involves finding plaintext values that produce a first intermediate value of
A = 0 (Figure 7.1b) and then using the meet-in-the-middle attack to determine
the two keys. The level of effort is 256, but the technique requires 256 chosen plain-
text–ciphertext pairs, which is a number unlikely to be provided by the holder of
the keys.

A known-plaintext attack is outlined in [VANO90]. This method is an im-
provement over the chosen-plaintext approach but requires more effort. The attack
is based on the observation that if we know A and C (Figure 7.1b), then the problem
reduces to that of an attack on double DES. Of course, the attacker does not know
A, even if P and C are known, as long as the two keys are unknown. However, the
attacker can choose a potential value of A and then try to find a known (P, C) pair
that produces A. The attack proceeds as follows.

1. Obtain n (P, C) pairs. This is the known plaintext. Place these in a table
(Table 1) sorted on the values of P (Figure 7.2b).

1American National Standards Institute (ANSI): Financial Institution Key Management (Wholesale).
From its title, X9.17 appears to be a somewhat obscure standard. Yet a number of techniques specified in
this standard have been adopted for use in other standards and applications, as we shall see throughout
this book.

212 CHAPTER 7 / BLOCK CIPHER OPERATION

2. Pick an arbitrary value a for A, and create a second table (Figure 7.2c) with en-
tries defined in the following fashion. For each of the 256 possible keys K1 = i,
calculate the plaintext value Pi such that

Pi = D(i, a)

For each Pi that matches an entry in Table 1, create an entry in Table 2 consist-
ing of the K1 value and the value of B that is produced for the (P, C) pair from
Table 1, assuming that value of K1:

B = D(i, C)

At the end of this step, sort Table 2 on the values of B.

3. We now have a number of candidate values of K1 in Table 2 and are in a
position to search for a value of K2. For each of the 2

56 possible keys K2 = j,
calculate the second intermediate value for our chosen value of a:

Bj = D(j, a)

At each step, look up Bj in Table 2. If there is a match, then the corresponding
key i from Table 2 plus this value of j are candidate values for the unknown
keys (K1, K2). Why? Because we have found a pair of keys (i, j) that produce a
known (P, C) pair (Figure 7.2a).

4. Test each candidate pair of keys (i, j) on a few other plaintext–ciphertext pairs.
If a pair of keys produces the desired ciphertext, the task is complete. If no pair
succeeds, repeat from step 1 with a new value of a.

Figure 7.2 Known-Plaintext Attack on Triple DES

E D E

i j i

Ci
a Bj

(a) Two-key triple encryption with candidate pair of keys

Pi

Pi Ci

(b) Table of n known
plaintext–ciphertext

pairs, sorted on P

Bj Key i

(c) Table of intermediate
values and candidate

keys

7.2 / ELECTRONIC CODEBOOK 213

For a given known (P, C), the probability of selecting the unique value of a
that leads to success is 1/264. Thus, given n (P, C) pairs, the probability of success for
a single selected value of a is n/264. A basic result from probability theory is that the
expected number of draws required to draw one red ball out of a bin containing n
red balls and N - n green balls is (N + 1)/(n + 1) if the balls are not replaced. So
the expected number of values of a that must be tried is, for large n,

264 + 1
n + 1

≈
264

n

Thus, the expected running time of the attack is on the order of

(256)
264

n
= 2120 - log2 n

Triple DES with Three Keys

Although the attacks just described appear impractical, anyone using two-key 3DES
may feel some concern. Thus, many researchers now feel that three-key 3DES is the
preferred alternative (e.g., [KALI96a]). In SP 800-57, Part 1 (Recommendation for
Key Management—Part 1: General, July 2012) NIST recommends that 2-key 3DES
be retired as soon as practical and replaced with 3-key 3DES.

Three-key 3DES is defined as

C = E(K3, D(K2, E(K1, P)))

Backward compatibility with DES is provided by putting K3 = K2 or K1 = K2. One
might expect that 3TDEA would provide 56 # 3 = 168 bits of strength. However,
there is an attack on 3TDEA that reduces the strength to the work that would be
involved in exhausting a 112-bit key [MERK81].

A number of Internet-based applications have adopted three-key 3DES, in-
cluding PGP and S/MIME, both discussed in Chapter 19.

7.2 ELECTRONIC CODEBOOK

A block cipher takes a fixed-length block of text of length b bits and a key as input
and produces a b-bit block of ciphertext. If the amount of plaintext to be encrypted
is greater than b bits, then the block cipher can still be used by breaking the plain-
text up into b-bit blocks. When multiple blocks of plaintext are encrypted using the
same key, a number of security issues arise. To apply a block cipher in a variety of
applications, five modes of operation have been defined by NIST (SP 800-38A).
In essence, a mode of operation is a technique for enhancing the effect of a cryp-
tographic algorithm or adapting the algorithm for an application, such as applying
a block cipher to a sequence of data blocks or a data stream. The five modes are
intended to cover a wide variety of applications of encryption for which a block
cipher could be used. These modes are intended for use with any symmetric block
cipher, including triple DES and AES. The modes are summarized in Table 7.1 and
described in this and the following sections.

214 CHAPTER 7 / BLOCK CIPHER OPERATION

The simplest mode is the electronic codebook (ECB) mode, in which plaintext
is handled one block at a time and each block of plaintext is encrypted using the
same key (Figure 7.3). The term codebook is used because, for a given key, there is
a unique ciphertext for every b-bit block of plaintext. Therefore, we can imagine a
gigantic codebook in which there is an entry for every possible b-bit plaintext pat-
tern showing its corresponding ciphertext.

For a message longer than b bits, the procedure is simply to break the message
into b-bit blocks, padding the last block if necessary. Decryption is performed one
block at a time, always using the same key. In Figure 7.3, the plaintext (padded as
necessary) consists of a sequence of b-bit blocks, P1, P2, c , PN; the correspond-
ing sequence of ciphertext blocks is C1, C2, c , CN. We can define ECB mode as
follows.

ECB C j = E(K, Pj) j = 1, c , N Pj = D(K, Cj) j = 1, c , N

The ECB mode should be used only to secure messages shorter than a single
block of underlying cipher (i.e., 64 bits for 3DES and 128 bits for AES), such as to
encrypt a secret key. Because in most of the cases messages are longer than the en-
cryption block mode, this mode has a minimum practical value.

The most significant characteristic of ECB is that if the same b-bit block of
plaintext appears more than once in the message, it always produces the same
ciphertext.

Mode Description Typical Application

Electronic Codebook (ECB) Each block of plaintext bits is
encoded independently using the
same key.

Secure transmission of
single values (e.g., an
encryption key)

Cipher Block Chaining (CBC) The input to the encryption algo-
rithm is the XOR of the next block
of plaintext and the preceding
block of ciphertext.

General-purpose block-
oriented transmission
Authentication

Cipher Feedback (CFB) Input is processed s bits at a time.
Preceding ciphertext is used as
input to the encryption algorithm
to produce pseudorandom output,
which is XORed with plaintext to
produce next unit of ciphertext.

General-purpose
stream-oriented
transmission
Authentication

Output Feedback (OFB) Similar to CFB, except that the
input to the encryption algorithm
is the preceding encryption output,
and full blocks are used.

Stream-oriented
transmission over noisy
channel (e.g., satellite
communication)

Counter (CTR) Each block of plaintext is XORed
with an encrypted counter. The
counter is incremented for each
subsequent block.

General-purpose block-
oriented transmission
Useful for high-speed
requirements

Table 7.1 Block Cipher Modes of Operation

7.2 / ELECTRONIC CODEBOOK 215

For lengthy messages, the ECB mode may not be secure. If the message is
highly structured, it may be possible for a cryptanalyst to exploit these regularities.
For example, if it is known that the message always starts out with certain predefined
fields, then the cryptanalyst may have a number of known plaintext–ciphertext pairs
to work with. If the message has repetitive elements with a period of repetition a
multiple of b bits, then these elements can be identified by the analyst. This may help
in the analysis or may provide an opportunity for substituting or rearranging blocks.

We now turn to more complex modes of operation. [KNUD00] lists the fol-
lowing criteria and properties for evaluating and constructing block cipher modes of
operation that are superior to ECB:

■ Overhead: The additional operations for the encryption and decryption opera-
tion when compared to encrypting and decrypting in the ECB mode.

■ Error recovery: The property that an error in the ith ciphertext block is inher-
ited by only a few plaintext blocks after which the mode resynchronizes.

■ Error propagation: The property that an error in the ith ciphertext block is
inherited by the ith and all subsequent plaintext blocks. What is meant here is
a bit error that occurs in the transmission of a ciphertext block, not a computa-
tional error in the encryption of a plaintext block.

Figure 7.3 Electronic Codebook (ECB) Mode

C1

P1

Encrypt

K

P2

C2

Encrypt

K

P N

CN

Encrypt

K

(a) Encryption

P1

C1

Decrypt

K

C2

P2

Decrypt

K

CN

PN

Decrypt

K

(b) Decryption

216 CHAPTER 7 / BLOCK CIPHER OPERATION

■ Diffusion: How the plaintext statistics are reflected in the ciphertext. Low en-
tropy plaintext blocks should not be reflected in the ciphertext blocks. Roughly,
low entropy equates to predictability or lack of randomness (see Appendix F).

■ Security: Whether or not the ciphertext blocks leak information about the
plaintext blocks.

7.3 CIPHER BLOCK CHAINING MODE

To overcome the security deficiencies of ECB, we would like a technique in which
the same plaintext block, if repeated, produces different ciphertext blocks. A
simple way to satisfy this requirement is the cipher block chaining (CBC) mode
(Figure 7.4). In this scheme, the input to the encryption algorithm is the XOR of the
current plaintext block and the preceding ciphertext block; the same key is used for
each block. In effect, we have chained together the processing of the sequence of
plaintext blocks. The input to the encryption function for each plaintext block bears
no fixed relationship to the plaintext block. Therefore, repeating patterns of b bits
are not exposed. As with the ECB mode, the CBC mode requires that the last block
be padded to a full b bits if it is a partial block.

Figure 7.4 Cipher Block Chaining (CBC) Mode

C1

P1

Encrypt

IV

K

P2

C2

Encrypt

K

PN

CN

CN–1

Encrypt

K

(a) Encryption

P1

C1

Decrypt

IV

K

C2

P2

Decrypt

K

CN

PN

CN–1

Decrypt

K

(b) Decryption

7.3 / CIPHER BLOCK CHAINING MODE 217

For decryption, each cipher block is passed through the decryption algorithm.
The result is XORed with the preceding ciphertext block to produce the plaintext
block. To see that this works, we can write

Cj = E(K, [Cj- 1⊕ Pj])

Then

D(K, Cj) = D(K, E(K, [Cj- 1⊕ Pj]))
D(K, Cj) = Cj- 1⊕ Pj

Cj- 1⊕D(K, Cj) = Cj- 1⊕ Cj- 1⊕ Pj = Pj

To produce the first block of ciphertext, an initialization vector (IV) is XORed
with the first block of plaintext. On decryption, the IV is XORed with the output
of the decryption algorithm to recover the first block of plaintext. The IV is a data
block that is the same size as the cipher block. We can define CBC mode as

CBC
C1 = E(K, [P1⊕ IV])

Cj = E(K, [Pj⊕ Cj- 1])j = 2, c , N

P1 = D(K, C1)⊕ IV

Pj = D(K, Cj)⊕ Cj- 1 j = 2, c , N

The IV must be known to both the sender and receiver but be unpredictable
by a third party. In particular, for any given plaintext, it must not be possible to
predict the IV that will be associated to the plaintext in advance of the generation
of the IV. For maximum security, the IV should be protected against unauthorized
changes. This could be done by sending the IV using ECB encryption. One reason
for protecting the IV is as follows: If an opponent is able to fool the receiver into
using a different value for IV, then the opponent is able to invert selected bits in the
first block of plaintext. To see this, consider

C1 = E(K, [IV⊕ P1])
P1 = IV⊕D(K, C1)

Now use the notation that X[i] denotes the ith bit of the b-bit quantity X. Then

P1[i] = IV[i]⊕D(K, C1)[i]

Then, using the properties of XOR, we can state

P1[i]′ = IV[i]′ ⊕D(K, C1)[i]

where the prime notation denotes bit complementation. This means that if an oppo-
nent can predictably change bits in IV, the corresponding bits of the received value
of P1 can be changed.

For other possible attacks based on prior knowledge of IV, see [VOYD83].
So long as it is unpredictable, the specific choice of IV is unimportant.

SP 800-38A recommends two possible methods: The first method is to apply
the encryption function, under the same key that is used for the encryption of the
plaintext, to a nonce.2 The nonce must be a data block that is unique to each

2NIST SP 800-90 (Recommendation for Random Number Generation Using Deterministic Random Bit
Generators) defines nonce as follows: A time-varying value that has at most a negligible chance of repeat-
ing, for example, a random value that is generated anew for each use, a timestamp, a sequence number,
or some combination of these.

Hiva-Network.Com

218 CHAPTER 7 / BLOCK CIPHER OPERATION

execution of the encryption operation. For example, the nonce may be a counter,
a timestamp, or a message number. The second method is to generate a random
data block using a random number generator.

In conclusion, because of the chaining mechanism of CBC, it is an appropriate
mode for encrypting messages of length greater than b bits.

In addition to its use to achieve confidentiality, the CBC mode can be used for
authentication. This use is described in Chapter 12.

7.4 CIPHER FEEDBACK MODE

For AES, DES, or any block cipher, encryption is performed on a block of b bits.
In the case of DES, b = 64 and in the case of AES, b = 128. However, it is pos-
sible to convert a block cipher into a stream cipher, using one of the three modes
to be discussed in this and the next two sections: cipher feedback (CFB) mode,
output feedback (OFB) mode, and counter (CTR) mode. A stream cipher elimi-
nates the need to pad a message to be an integral number of blocks. It also can
operate in real time. Thus, if a character stream is being transmitted, each char-
acter can be encrypted and transmitted immediately using a character-oriented
stream cipher.

One desirable property of a stream cipher is that the ciphertext be of the same
length as the plaintext. Thus, if 8-bit characters are being transmitted, each charac-
ter should be encrypted to produce a ciphertext output of 8 bits. If more than 8 bits
are produced, transmission capacity is wasted.

Figure 7.5 depicts the CFB scheme. In the figure, it is assumed that the unit of
transmission is s bits; a common value is s = 8. As with CBC, the units of plaintext
are chained together, so that the ciphertext of any plaintext unit is a function of all
the preceding plaintext. In this case, rather than blocks of b bits, the plaintext is
divided into segments of s bits.

First, consider encryption. The input to the encryption function is a b-bit shift
register that is initially set to some initialization vector (IV). The leftmost (most
significant) s bits of the output of the encryption function are XORed with the first
segment of plaintext P1 to produce the first unit of ciphertext C1, which is then
transmitted. In addition, the contents of the shift register are shifted left by s bits,
and C1 is placed in the rightmost (least significant) s bits of the shift register. This
process continues until all plaintext units have been encrypted.

For decryption, the same scheme is used, except that the received ciphertext
unit is XORed with the output of the encryption function to produce the plaintext
unit. Note that it is the encryption function that is used, not the decryption function.
This is easily explained. Let MSBs(X) be defined as the most significant s bits of X.
Then

C1 = P1⊕MSBs[E(K, IV)]

Therefore, by rearranging terms:

P1 = C1⊕MSBs[E(K, IV)]

The same reasoning holds for subsequent steps in the process.

7.4 / CIPHER FEEDBACK MODE 219

We can define CFB mode as follows.

CFB

I1 = IV

Ij = LSBb - s(Ij- 1) }Cj- 1 j = 2, c , N

Oj = E(K, Ij) j = 1, c , N

Cj = Pj⊕MSBs(Oj) j = 1, c , N

I1 = IV

Ij = LSBb - s(Ij- 1) }Cj- 1 j = 2, c , N

Oj = E(K, Ij) j = 1, c , N

Pj = Cj⊕MSBs(Oj) j = 1, c , N

Although CFB can be viewed as a stream cipher, it does not conform to the
typical construction of a stream cipher. In a typical stream cipher, the cipher takes

Figure 7.5 s-bit Cipher Feedback (CFB) Mode

C1

IV
I1

O1

I1

O1

I2

O2

I2

O2

IN

ON

IN

ON

P1

Encrypt

Select
s bits

Discard
b – s bits

K

(a) Encryption

CN–1

(b) Decryption

s bits

s bits s bits

C2

P2

Encrypt

Select
s bits

Discard
b – s bits

K

s bits

s bitsb – s bits
Shift register

s bits

CN

PN

Encrypt

Select
s bits

Discard
b – s bits

K

s bits

s bitsb – s bits
Shift register

P1

IV

C1

Encrypt

Select
s bits

Discard
b – s bits

K

CN–1

s bits
C2

s bits
CN

s bits

s bits s bits

P2

Encrypt

Select
s bits

Discard
b – s bits

K
s bitsb – s bits

Shift register
s bitsb – s bits

Shift register

s bits

PN

Encrypt

Select
s bits

Discard
b – s bits

K

220 CHAPTER 7 / BLOCK CIPHER OPERATION

as input some initial value and a key and generates a stream of bits, which is then
XORed with the plaintext bits (see Figure 4.1). In the case of CFB, the stream of
bits that is XORed with the plaintext also depends on the plaintext.

In CFB encryption, like CBC encryption, the input block to each forward
cipher function (except the first) depends on the result of the previous forward
cipher function; therefore, multiple forward cipher operations cannot be performed
in parallel. In CFB decryption, the required forward cipher operations can be per-
formed in parallel if the input blocks are first constructed (in series) from the IV
and the ciphertext.

7.5 OUTPUT FEEDBACK MODE

The output feedback (OFB) mode is similar in structure to that of CFB. For OFB,
the output of the encryption function is fed back to become the input for encrypting
the next block of plaintext (Figure 7.6). In CFB, the output of the XOR unit is fed
back to become input for encrypting the next block. The other difference is that the
OFB mode operates on full blocks of plaintext and ciphertext, whereas CFB oper-
ates on an s-bit subset. OFB encryption can be expressed as

Cj = Pj⊕ E(K, Oj- 1)

where

Oj- 1 = E(K, Oj- 2)

Some thought should convince you that we can rewrite the encryption expres-
sion as:

Cj = Pj⊕ E(K, [Cj- 1⊕ Pj- 1])

By rearranging terms, we can demonstrate that decryption works.

Pj = Cj⊕ E(K, [Cj- 1⊕ Pj- 1])

We can define OFB mode as follows.

OFB

I1 = Nonce

Ij = Oj- 1 j = 2, c , N

Oj = E(K, Ij) j = 1, c , N

Cj = Pj⊕ Oj j = 1, c , N - 1
CN
* = PN* ⊕MSBu(ON)

I1 = Nonce

Ij = Oj- 1 j = 2, c , N

Oj = E(K, Ij) j = 1, c , N

Pj = Cj⊕ Oj j = 1, c , N - 1
PN
* = CN* ⊕MSBu(ON)

Let the size of a block be b. If the last block of plaintext contains u bits (indi-
cated by *), with u 6 b, the most significant u bits of the last output block ON are
used for the XOR operation; the remaining b - u bits of the last output block are
discarded.

As with CBC and CFB, the OFB mode requires an initialization vector. In
the case of OFB, the IV must be a nonce; that is, the IV must be unique to each
execution of the encryption operation. The reason for this is that the sequence of

7.5 / OUTPUT FEEDBACK MODE 221

encryption output blocks, Oi, depends only on the key and the IV and does not de-
pend on the plaintext. Therefore, for a given key and IV, the stream of output bits
used to XOR with the stream of plaintext bits is fixed. If two different messages had
an identical block of plaintext in the identical position, then an attacker would be
able to determine that portion of the Oi stream.

One advantage of the OFB method is that bit errors in transmission do not
propagate. For example, if a bit error occurs in C1, only the recovered value of P1 is
affected; subsequent plaintext units are not corrupted. With CFB, C1 also serves as
input to the shift register and therefore causes additional corruption downstream.

The disadvantage of OFB is that it is more vulnerable to a message stream
modification attack than is CFB. Consider that complementing a bit in the cipher-
text complements the corresponding bit in the recovered plaintext. Thus, controlled

Figure 7.6 Output Feedback (OFB) Mode

(a) Encryption

P1

C1

Nonce

Encrypt

K

P2 PN

C2

Encrypt

K

CN

Encrypt

K

(b) Decryption

C1

I1 I2 IN

I1 I2 IN

O1 O2 ON

O1 O2 ON

P1

Nonce

Encrypt

K

C2 CN

P2

Encrypt

K

PN

Encrypt

K

222 CHAPTER 7 / BLOCK CIPHER OPERATION

changes to the recovered plaintext can be made. This may make it possible for an
opponent, by making the necessary changes to the checksum portion of the message
as well as to the data portion, to alter the ciphertext in such a way that it is not de-
tected by an error-correcting code. For a further discussion, see [VOYD83].

OFB has the structure of a typical stream cipher, because the cipher gener-
ates a stream of bits as a function of an initial value and a key, and that stream of
bits is XORed with the plaintext bits (see Figure 4.1). The generated stream that is
XORed with the plaintext is itself independent of the plaintext; this is highlighted
by dashed boxes in Figure 7.6. One distinction from the stream ciphers we discuss
in Chapter 8 is that OFB encrypts plaintext a full block at a time, where typically a
block is 64 or 128 bits. Many stream ciphers encrypt one byte at a time.

7.6 COUNTER MODE

Although interest in the counter (CTR) mode has increased recently with appli-
cations to ATM (asynchronous transfer mode) network security and IPsec
(IP security), this mode was proposed in 1979 (e.g., [DIFF79]).

Figure 7.7 depicts the CTR mode. A counter equal to the plaintext block size
is used. The only requirement stated in SP 800-38A is that the counter value must be
different for each plaintext block that is encrypted. Typically, the counter is initial-
ized to some value and then incremented by 1 for each subsequent block (modulo 2b,
where b is the block size). For encryption, the counter is encrypted and then XORed
with the plaintext block to produce the ciphertext block; there is no chaining. For
decryption, the same sequence of counter values is used, with each encrypted coun-
ter XORed with a ciphertext block to recover the corresponding plaintext block.
Thus, the initial counter value must be made available for decryption. Given a
sequence of counters T1, T2, c , TN, we can define CTR mode as follows.

CTR
Cj = Pj⊕ E(K, Tj) j = 1, c , N - 1

CN
* = PN* ⊕MSBu[E(K, TN)]

Pj = Cj⊕ E(K, Tj) j = 1, c , N - 1

PN
* = CN* ⊕MSBu[E(K, TN)]

For the last plaintext block, which may be a partial block of u bits, the most
significant u bits of the last output block are used for the XOR operation; the re-
maining b - u bits are discarded. Unlike the ECB, CBC, and CFB modes, we do
not need to use padding because of the structure of the CTR mode.

As with the OFB mode, the initial counter value must be a nonce; that is, T1
must be different for all of the messages encrypted using the same key. Further,
all Ti values across all messages must be unique. If, contrary to this requirement, a
counter value is used multiple times, then the confidentiality of all of the plaintext
blocks corresponding to that counter value may be compromised. In particular, if
any plaintext block that is encrypted using a given counter value is known, then
the output of the encryption function can be determined easily from the associated
ciphertext block. This output allows any other plaintext blocks that are encrypted
using the same counter value to be easily recovered from their associated ciphertext
blocks.

7.6 / COUNTER MODE 223

One way to ensure the uniqueness of counter values is to continue to incre-
ment the counter value by 1 across messages. That is, the first counter value of the
each message is one more than the last counter value of the preceding message.

[LIPM00] lists the following advantages of CTR mode.

■ Hardware efficiency: Unlike the three chaining modes, encryption (or decryp-
tion) in CTR mode can be done in parallel on multiple blocks of plaintext or
ciphertext. For the chaining modes, the algorithm must complete the computa-
tion on one block before beginning on the next block. This limits the maximum
throughput of the algorithm to the reciprocal of the time for one execution of
block encryption or decryption. In CTR mode, the throughput is only limited
by the amount of parallelism that is achieved.

Figure 7.7 Counter (CTR) Mode

(a) Encryption

P1

C1

Counter 1

Encrypt

K

Counter 2 Counter N

P2 PN

C2

Encrypt

K

CN

Encrypt

K

(b) Decryption

C1

P1

Counter 1

Encrypt

K

Counter 2 Counter N

C2 CN

P2

Encrypt

K

PN

Encrypt

K

224 CHAPTER 7 / BLOCK CIPHER OPERATION

■ Software efficiency: Similarly, because of the opportunities for parallel execu-
tion in CTR mode, processors that support parallel features, such as aggressive
pipelining, multiple instruction dispatch per clock cycle, a large number of reg-
isters, and SIMD instructions, can be effectively utilized.

■ Preprocessing: The execution of the underlying encryption algorithm does
not depend on input of the plaintext or ciphertext. Therefore, if sufficient
memory is available and security is maintained, preprocessing can be used to
prepare the output of the encryption boxes that feed into the XOR functions,
as in Figure 7.7. When the plaintext or ciphertext input is presented, then
the only computation is a series of XORs. Such a strategy greatly enhances
throughput.

■ Random access: The ith block of plaintext or ciphertext can be processed in
random-access fashion. With the chaining modes, block Ci cannot be com-
puted until the i - 1 prior blocks are computed. There may be applications in
which a ciphertext is stored and it is desired to decrypt just one block; for such
applications, the random access feature is attractive.

■ Provable security: It can be shown that CTR is at least as secure as the other
modes discussed in this chapter.

■ Simplicity: Unlike ECB and CBC modes, CTR mode requires only the imple-
mentation of the encryption algorithm and not the decryption algorithm. This
matters most when the decryption algorithm differs substantially from the en-
cryption algorithm, as it does for AES. In addition, the decryption key schedul-
ing need not be implemented.

Note that, with the exception of ECB, all of the NIST-approved block ci-
pher modes of operation involve feedback. This is clearly seen in Figure 7.8. To
highlight the feedback mechanism, it is useful to think of the encryption function
as taking input from an input register whose length equals the encryption block
length and with output stored in an output register. The input register is updated
one block at a time by the feedback mechanism. After each update, the encryp-
tion algorithm is executed, producing a result in the output register. Meanwhile,
a block of plaintext is accessed. Note that both OFB and CTR produce output
that is independent of both the plaintext and the ciphertext. Thus, they are natu-
ral candidates for stream ciphers that encrypt plaintext by XOR one full block at
a time.

7.7 XTS-AES MODE FOR BLOCK-ORIENTED
STORAGE DEVICES

In 2010, NIST approved an additional block cipher mode of operation, XTS-AES.
This mode is also an IEEE standard, IEEE Std 1619-2007, which was developed
by the IEEE Security in Storage Working Group (P1619). The standard describes
a method of encryption for data stored in sector-based devices where the threat
model includes possible access to stored data by the adversary. The standard has
received widespread industry support.

7.7 / XTS-AES MODE FOR BLOCK-ORIENTED STORAGE DEVICES 225

Tweakable Block Ciphers

The XTS-AES mode is based on the concept of a tweakable block cipher, intro-
duced in [LISK02], which functions in much the same manner as a salt used with
passwords, as described in Chapter 22. The form of this concept used in XTS-AES
was first described in [ROGA04].

Before examining XTS-AES, let us consider the general structure of a tweak-
able block cipher. A tweakable block cipher is one that has three inputs: a plain-
text P, a symmetric key K, and a tweak T; and produces a ciphertext output C. We
can write this as C = E(K, T, P). The tweak need not be kept secret. Whereas the

Figure 7.8 Feedback Characteristic of Modes of Operation

Plaintext block

Plaintext block

Encrypt

Input register

Output register

Ciphertext Ciphertext

(a) Cipher block chaining (CBC) mode

Key

Encrypt

Input register

Output register

Key

(b) Cipher feedback (CFB) mode

Plaintext block

Ciphertext

Key

Encrypt

Input register

Output register

(c) Output feedback (OFB) mode

Plaintext block

Ciphertext

Key

Encrypt

Input register

Output register

Counter

(d) Counter (CTR) mode

226 CHAPTER 7 / BLOCK CIPHER OPERATION

purpose of the key is to provide security, the purpose of the tweak is to provide
variability. That is, the use of different tweaks with the same plaintext and same key
produces different outputs. The basic structure of several tweakable clock ciphers
that have been implemented is shown in Figure 7.9. Encryption can be expressed as:

C = H(T)⊕ E(K, H(T)⊕ P)

where H is a hash function. For decryption, the same structure is used with the
plaintext as input and decryption as the function instead of encryption. To see that
this works, we can write

H(T)⊕ C = E(K, H(T)⊕ P)
D[K, H(T)⊕ C] = H(T)⊕ P
H(T)⊕D(K, H(T)⊕ C) = P

It is now easy to construct a block cipher mode of operation by using a differ-
ent tweak value on each block. In essence, the ECB mode is used but for each block
the tweak is changed. This overcomes the principal security weakness of ECB,
which is that two encryptions of the same block yield the same ciphertext.

Storage Encryption Requirements

The requirements for encrypting stored data, also referred to as “data at rest” dif-
fer somewhat from those for transmitted data. The P1619 standard was designed to
have the following characteristics:

1. The ciphertext is freely available for an attacker. Among the circumstances
that lead to this situation:

a. A group of users has authorized access to a database. Some of the records in
the database are encrypted so that only specific users can successfully read/

Figure 7.9 Tweakable Block Cipher

K

Hash
function

Tj

H(Tj)

Pj

Cj

Encrypt

(a) Encryption

K

Hash
function

Tj Cj

Pj

Decrypt

(b) Decryption

Hiva-Network.Com

7.7 / XTS-AES MODE FOR BLOCK-ORIENTED STORAGE DEVICES 227

write them. Other users can retrieve an encrypted record but are unable to
read it without the key.

b. An unauthorized user manages to gain access to encrypted records.
c. A data disk or laptop is stolen, giving the adversary access to the encrypted

data.
2. The data layout is not changed on the storage medium and in transit. The en-

crypted data must be the same size as the plaintext data.

3. Data are accessed in fixed sized blocks, independently from each other. That is,
an authorized user may access one or more blocks in any order.

4. Encryption is performed in 16-byte blocks, independently from other blocks
(except the last two plaintext blocks of a sector, if its size is not a multiple of
16 bytes).

5. There are no other metadata used, except the location of the data blocks
within the whole data set.

6. The same plaintext is encrypted to different ciphertexts at different locations,
but always to the same ciphertext when written to the same location again.

7. A standard conformant device can be constructed for decryption of data en-
crypted by another standard conformant device.

The P1619 group considered some of the existing modes of operation for use with
stored data. For CTR mode, an adversary with write access to the encrypted media can
flip any bit of the plaintext simply by flipping the corresponding ciphertext bit.

Next, consider requirement 6 and the use of CBC. To enforce the requirement
that the same plaintext encrypts to different ciphertext in different locations, the IV
could be derived from the sector number. Each sector contains multiple blocks. An
adversary with read/write access to the encrypted disk can copy a ciphertext sec-
tor from one position to another, and an application reading the sector off the new
location will still get the same plaintext sector (except perhaps the first 128 bits).
For example, this means that an adversary that is allowed to read a sector from the
second position but not the first can find the content of the sector in the first posi-
tion by manipulating the ciphertext. Another weakness is that an adversary can flip
any bit of the plaintext by flipping the corresponding ciphertext bit of the previous
block, with the side-effect of “randomizing” the previous block.

Operation on a Single Block

Figure 7.10 shows the encryption and decryption of a single block. The operation in-
volves two instances of the AES algorithm with two keys. The following parameters
are associated with the algorithm.

Key The 256 or 512 bit XTS-AES key; this is parsed as a concatenation of two
fields of equal size called Key1 and Key2, such that Key = Key1 }Key2 .

Pj The jth block of plaintext. All blocks except possibly the final block have a
length of 128 bits. A plaintext data unit, typically a disk sector, consists of a
sequence of plaintext blocks P1, P2, c , Pm.

Cj The jth block of ciphertext. All blocks except possibly the final block have a
length of 128 bits.

228 CHAPTER 7 / BLOCK CIPHER OPERATION

j The sequential number of the 128-bit block inside the data unit.

i The value of the 128-bit tweak. Each data unit (sector) is assigned a
tweak value that is a nonnegative integer. The tweak values are assigned
consecutively, starting from an arbitrary nonnegative integer.

a A primitive element of GF(2128) that corresponds to polynomial x
(i.e., 0000c 0102).

aj a multiplied by itself j times, in GF(2128).

⊕ Bitwise XOR.

⊗ Modular multiplication of two polynomials with binary coefficients modulo
x128 + x7 + x2 + x + 1. Thus, this is multiplication in GF(2128).

Figure 7.10 XTS-AES Operation on Single Block

Key2

Key1

AES
Encrypt

i

T

CC

PP

Pj

Cj

AES
Encrypt

(a) Encryption

(b) Decryption

j

Key2

Key1

AES
Encrypt

i

T

CC

PP

Cj

Pj

AES
Decrypt

j

7.7 / XTS-AES MODE FOR BLOCK-ORIENTED STORAGE DEVICES 229

In essence, the parameter j functions much like the counter in CTR mode. It
assures that if the same plaintext block appears at two different positions within a
data unit, it will encrypt to two different ciphertext blocks. The parameter i functions
much like a nonce at the data unit level. It assures that, if the same plaintext block
appears at the same position in two different data units, it will encrypt to two differ-
ent ciphertext blocks. More generally, it assures that the same plaintext data unit will
encrypt to two different ciphertext data units for two different data unit positions.

The encryption and decryption of a single block can be described as

XTS-AES block
operation

T = E(K2, i)⊗ aj

PP = P⊕ T
CC = E(K1, PP)

C = CC⊕ T

T = E(K2, i)⊗ aj

CC = C⊕ T
PP = D(K1, CC)

P = PP⊕ T

To see that decryption recovers the plaintext, let us expand the last line of both en-
cryption and decryption. For encryption, we have

C = CC⊕ T = E(K1, PP)⊕ T = E(K1, P⊕ T)⊕ T

and for decryption, we have

P = PP⊕ T = D(K1, CC)⊕ T = D(K1, C⊕ T)⊕ T

Now, we substitute for C:

P = D(K1, C⊕ T)⊕ T
= D(K1, [E(K1, P⊕ T)⊕ T]⊕ T)⊕ T
= D(K1, E(K1, P⊕ T))⊕ T
= (P⊕ T)⊕ T = P

Operation on a Sector

The plaintext of a sector or data unit is organized into blocks of 128 bits. Blocks are
labeled P0, P1, c , Pm. The last block my be null or may contain from 1 to 127 bits.
In other words, the input to the XTS-AES algorithm consists of m 128-bit blocks
and possibly a final partial block.

For encryption and decryption, each block is treated independently and en-
crypted/decrypted as shown in Figure 7.10. The only exception occurs when the
last block has less than 128 bits. In that case, the last two blocks are encrypted/de-
crypted using a ciphertext-stealing technique instead of padding. Figure 7.11 shows
the scheme. Pm - 1 is the last full plaintext block, and Pm is the final plaintext block,
which contains s bits with 1 … s … 127. Cm - 1 is the last full ciphertext block, and
Cm is the final ciphertext block, which contains s bits. This technique is commonly
called ciphertext stealing because the processing of the last block “steals” a tempo-
rary ciphertext of the penultimate block to complete the cipher block.

Let us label the block encryption and decryption algorithms of Figure 7.10 as

Block encryption: XTS-AES-blockEnc(K, Pj, i, j)
Block decryption: XTS-AES-blockDec(K, Cj, i, j)

230 CHAPTER 7 / BLOCK CIPHER OPERATION

Then, XTS-AES mode is defined as follows:

XTS-AES mode with null
final block

Cj = [email protected]@blockEnc(K, Pj, i, j) j = 0, c , m - 1

Pj = [email protected]@blockEnc(K, Cj, i, j) j = 0, c , m - 1

XTS-AES mode with final
block containing s bits

Cj = [email protected]@blockEnc(K, Pj, i, j) j = 0, c , m - 2
XX = [email protected]@blockEnc(K, Pm - 1, i, m - 1)
CP = LSB128 - s(XX)
YY = Pm }CP

Cm - 1 = [email protected]@blockEnc(K, YY, i, m)
Cm = MSBs(XX)

Pj = [email protected]@blockDec(K, Cj, i, j) j = 0, c , m - 2
YY = [email protected]@blockDec(K, Cm - 1, i, m - 1)
CP = LSB128 - s(YY)
XX = Cm }CP

Pm - 1 = [email protected]@blockDec(K, XX, i, m)
Pm = MSBs(YY)

Figure 7.11 XTS-AES Mode

C0

P0

XTS-AES
block

encryption

Key

i, 0

C1

P1

XTS-AES
block

encryption

Key

i, 1

CP

XX

XX

YY

YY

Cm

CPPmPm–1

XTS-AES
block

encryption

Key

i, m–1

Cm–1

Cm–1

XTS-AES
block

encryption

Key

i, m

Cm
(a) Encryption

(b) Decryption

P0

C0

XTS-AES
block

decryption

Key

i, 0

P1

C1

XTS-AES
block

decryption

Key

i, 1

CPPm

CPCmCm–1

XTS-AES
block

decryption

Key

i, m

Pm–1

Pm–1

XTS-AES
block

decryption

Key

i, m–1

Pm

7.8 / FORMAT-PRESERVING ENCRYPTION 231

As can be seen, XTS-AES mode, like CTR mode, is suitable for parallel oper-
ation. Because there is no chaining, multiple blocks can be encrypted or decrypted
simultaneously. Unlike CTR mode, XTS-AES mode includes a nonce (the param-
eter i) as well as a counter (parameter j).

7.8 FORMAT-PRESERVING ENCRYPTION

Format-preserving encryption (FPE) refers to any encryption technique that takes
a plaintext in a given format and produces a ciphertext in the same format. For
example, credit cards consist of 16 decimal digits. An FPE that can accept this type of
input would produce a ciphertext output of 16 decimal digits. Note that the ciphertext
need not be, and in fact is unlikely to be, a valid credit card number. But it will have
the same format and can be stored in the same way as credit card number plaintext.

A simple encryption algorithm is not format preserving, with the exception
that it preserves the format of binary strings. For example, Table 7.2 shows three
types of plaintext for which it might be desired to perform FPE. The third row
shows examples of what might be generated by an FPE algorithm. The fourth row
shows (in hexadecimal) what is produced by AES with a given key.

Motivation

FPE facilitates the retrofitting of encryption technology to legacy applications,
where a conventional encryption mode might not be feasible because it would dis-
rupt data fields/pathways. FPE has emerged as a useful cryptographic tool, whose
applications include financial-information security, data sanitization, and transpar-
ent encryption of fields in legacy databases.

The principal benefit of FPE is that it enables protection of particular data
elements in a legacy database that did not provide encryption of those data ele-
ments, while still enabling workflows that were in place before FPE was in use. With
FPE, as opposed to ordinary AES encryption or TDEA encryption, no database
schema changes and minimal application changes are required. Only applications
that need to see the plaintext of a data element need to be modified and generally
these modifications will be minimal.

Some examples of legacy applications where FPE is desirable:

■ COBOL data-processing applications: Any changes in the structure of a re-
cord requires corresponding changes in all code that references that record
structure. Typical code sizes involve hundreds of modules, each containing
around 5,000–10,000 lines on average.

Credit Card Tax ID Bank Account Number

Plaintext 8123 4512 3456 6780 219-09-9999 800N2982K-22

FPE 8123 4521 7292 6780 078-05-1120 709G9242H-35

AES (hex) af411326466add24
c86abd8aa525db7a

7b9af4f3f218ab25
07c7376869313afa

9720ec7f793096ff
d37141242e1c51bd

Table 7.2 Comparison of Format-Preserving Encryption and AES

232 CHAPTER 7 / BLOCK CIPHER OPERATION

■ Database applications: Fields that are specified to take only character strings
cannot be used to store conventionally encrypted binary ciphertext. Base64
encoding of such binary ciphertext is not always feasible without increase in
data lengths, requiring augmentation of corresponding field lengths.

■ FPE-encrypted characters can be significantly compressed for efficient trans-
mission. This cannot be said about AES-encrypted binary ciphertext.

Difficulties in Designing an FPE

A general-purpose standardized FPE should meet a number of requirements:

1. The ciphertext is of the same length and format as the plaintext.

2. It should be adaptable to work with a variety of character and number types.
Examples include decimal digits, lowercase alphabetic characters, and the full
character set of a standard keyboard or international keyboard.

3. It should work with variable plaintext lengths.

4. Security strength should be comparable to that achieved with AES.

5. Security should be strong even for very small plaintext lengths.

Meeting the first requirement is not at all straightforward. As illustrated in
Table 7.2, a straightforward encryption with AES yields a 128-bit binary block that
does not resemble the required format. Also, a standard symmetric block cipher is
not easily adaptable to produce an FPE.

Consider a simple example. Assume that we want an algorithm that can en-
crypt decimal digit strings of maximum length of 32 digits. The input to the algo-
rithm can be stored in 16 bytes (128 bits) by encoding each digit as four bits and
using the corresponding binary value for each digit (e.g., 6 is encoded as 0101).
Next, we use AES to encrypt the 128-bit block, in the following fashion:

1. The plaintext input X is represented by the string of 4-bit decimal digits
X[1] . . . X[16]. If the plaintext is less than 16 digits long, it is padded out to the
left (most significant) with zeros.

2. Treating X as a 128-bit binary string and using key K, form ciphertext
Y = AESK(X).

3. Treat Y as a string of length 16 of 4-bit elements.

4. Some of the entries in Y may have values greater than 9 (e.g., 1100). To gener-
ate ciphertext Z in the required format, calculate

Z[i] = Y[i] mod 10, 1 … i … 16

This generates a ciphertext of 16 decimal digits, which conforms to the de-
sired format. However, this algorithm does not meet the basic requirement of
any encryption algorithm of reversibility. It is impossible to decrypt Z to recover
the original plaintext X because the operation is one-way; that is, it is a many-
to-one function. For example, 12 mod 10 = 2 mod 10 = 2. Thus, we need to de-
sign a reversible function that is both a secure encryption algorithm and format
preserving.

7.8 / FORMAT-PRESERVING ENCRYPTION 233

A second difficulty in designing an FPE is that some of the input strings are
quite short. For example, consider the 16-digit credit card number (CCN). The first
six digits provide the issuer identification number (IIN), which identifies the insti-
tution that issued the card. The final digit is a check digit to catch typographical
errors or other mistakes. The remaining nine digits are the user’s account number.
However, a number of applications require that the last four digits be in the clear
(the check digit plus three account digits) for applications such as credit card re-
ceipts, which leaves only six digits for encryption. Now suppose that an adversary
is able to obtain a number of plaintext/ciphertext pairs. Each such pair corresponds
to not just one CCN, but multiple CCNs that have the same middle six digits. In a
large database of credit card numbers, there may be multiple card numbers with
the same middle six digits. An adversary may be able to assemble a large diction-
ary mapping known as six-digit plaintexts to their corresponding ciphertexts. This
could be used to decrypt unknown ciphertexts from the database. As pointed out
in [BELL10a], in a database of 100 million entries, on average about 100 CCNs
will share any given middle-six digits. Thus, if the adversary has learned k CCNs
and gains access to such a database, the adversary can decrypt approximately
100k CCNs.

The solution to this second difficulty is to use a tweakable block cipher; this
concept is described in Section 7.7. For example, the tweak for CCNs could be the
first two and last four digits of the CCN. Prior to encryption, the tweak is added,
digit-by-digit mod 10, to the middle six-digit plaintext, and the result is then en-
crypted. Two different CCNs with identical middle six digits will yield different
tweaked inputs and therefore different ciphertexts. Consider the following:

CCN Tweak Plaintext Plaintext + Tweak

4012 8812 3456 1884 401884 123456 524230

5105 1012 3456 6782 516782 123456 639138

Two CCNs with the same middle six digits have different tweaks and there-
fore different values to the middle six digits after the tweak is added.

Feistel Structure for Format-Preserving Encryption

As the preceding discussion shows, the challenge with FPE is to design an algo-
rithm for scrambling the plaintext that is secure, preserves format, and is reversible.
A number of approaches have been proposed in recent years [ROGA10, BELL09]
for FPE algorithms. The majority of these proposals use a Feistel structure.
Although IBM introduced this structure with their Lucifer cipher [SMIT71] almost
half a century ago, it remains a powerful basis for implementing ciphers.

This section provides a general description of how the Feistel structure can
be used to implement an FPE. In the following section, we look at three specific
Feistel-based algorithms that are in the process of receiving NIST approval.

ENCRYPTION AND DECRYPTION Figure 7.12 shows the Feistel structure used in all of
the NIST algorithms, with encryption shown on the left-hand side and decryption
on the right-hand side. The structure in Figure 7.12 is the same as that shown in

234 CHAPTER 7 / BLOCK CIPHER OPERATION

Figure 4.3 but, to simplify the presentation, it is untwisted, not illustrating the swap
that occurs at the end of each round.

The input to the encryption algorithm is a plaintext character string of
n = u + v characters. If n is even, then u = v, otherwise u and v differ by 1. The
two parts of the string pass through an even number of rounds of processing to
produce a ciphertext block of n characters and the same format as the plaintext.
Each round i has inputs Ai and Bi, derived from the preceding round (or plaintext
for round 0).

All rounds have the same structure. On even-numbered rounds, a substitution
is performed on the left part (length u) of the data, Ai. This is done by applying the
round function FK to the right part (length v) of the data, Bi, and then performing

Figure 7.12 Feistel Structure for Format-Preserving Encryption

Input (plaintext)

Output (ciphertext)

(a) Encryption (b) Decryption

R
ou

nd
0

R
ou

nd
1

A0

C0

C1

u characters v characters
B0

n, T, 0

n, T, 1

A2 B1 B2 C1

+ FK

+

B1 C0 A1 B0

FK

R
ou

nd
r–

2
R

ou
nd

r–
1

Ar–2

Cr–2

Br–2

n, T, r–2

n, T, r–1

Ar Br–1 Br Cr–1

+ FK

+

Br–1 Cr–2 Ar–1 Br–2

FK

Output (plaintext)

Input (ciphertext)

R
ou

nd
r–

1
R

ou
nd

r–
2

A0 C0

C0

C1

u characters v characters
B0 A1

n, T, 0

n, T, 1

A2 C2 B2 A3

– FK

–

B1 A2 A1 C1

FK

R
ou

nd
1

R
ou

nd
0

Ci–2

Cr–1

Cr–1

n, T, i–2

n, T, r–1

Ar Br

– FK

–

Br–1 Ar Ar–1 Cr–1

Ar–2 Cr–2 Br–2 Ar–1

FK

7.8 / FORMAT-PRESERVING ENCRYPTION 235

a modular addition of the output of FK with Ai. The modular addition function and
the selection of modulus are described subsequently. On odd-numbered rounds,
the substitution is done on the right part of the data. FK is a one-way function that
converts the input into a binary string, performs a scrambling transformation on the
string, and then converts the result back into a character string of suitable format
and length. The function has as parameters the secret key K, the plaintext length n,
a tweak T, and the round number i.

Note that on even-numbered rounds, FK has an input of v characters, and that
the modular addition produces a result of u characters, whereas on odd-numbered
rounds, FK has an input of u characters, and that the modular addition produces a
result of v characters. The total number of rounds is even, so that the output consists
of an A portion of length u concatenated with a B portion of length v, matching the
partition of the plaintext.

The process of decryption is essentially the same as the encryption process.
The differences are: (1) the addition function is replaced by a subtraction function
that is its inverse; and (2) the order of the round indices is reversed.

To demonstrate that the decryption produces the correct result, Figure 7.12b
shows the encryption process going down the left-hand side and the decryption pro-
cess going up the right-hand side. The diagram indicates that, at every round, the
intermediate value of the decryption process is equal to the corresponding value of
the encryption process. We can walk through the figure to validate this, starting at
the bottom. The ciphertext is produced at the end of round r - 1 as a string of the
form A

r }B r, with Ar and Br having string lengths u and v, respectively. Encryption
round r - 1 can be described with the following equations:

Ar = Br - 1
Br = Ar - 1 + FK[Br - 1]

Because we define the subtraction function to be the inverse of the addition
function, these equations can be rewritten:

Br - 1 = Ar
Ar - 1 = Br - FK[Br - 1]

It can be seen that the last two equations describe the action of round 0 of the
decryption function, so that the output of round 0 of decryption equals the input
of round r - 1 of encryption. This correspondence holds all the way through the r
iterations, as is easily shown.

Note that the derivation does not require that F be a reversible function. To
see this, take a limiting case in which F produces a constant output (e.g., all ones)
regardless of the values of its input. The equations still hold.

CHARACTER STRINGS The NIST algorithms, and the other FPE algorithms that have
been proposed, are used with plaintext consisting of a string of elements, called
characters. Specifically, a finite set of two or more symbols is called an alphabet,
and the elements of an alphabet are called characters. A character string is a finite
sequence of characters from an alphabet. Individual characters may repeat in the
string. The number of different characters in an alphabet is called the base, also

Hiva-Network.Com

236 CHAPTER 7 / BLOCK CIPHER OPERATION

referred to as the radix of the alphabet. For example, the lowercase English alpha-
bet a, b, c, . . . has a radix, or base, of 26. For purposes of encryption and decryption,
the plaintext alphabet must be converted to numerals, where a numeral is a non-
negative integer that is less than the base. For example, for the lowercase alphabet,
the assignment could be characters a, b, c, . . . , z map into 0, 1, 2, . . . , 25.

A limitation of this approach is that all of the elements in a plaintext format
must have the same radix. So, for example, an identification number that consists
of an alphabetic character followed by nine numeric digits cannot be handled in
format-preserving fashion by the FPEs that have been implemented so far.

The NIST document defines notation for specifying these conversions
(Table 7.3a). To begin, it is assumed that the character string is represented by
a numeral string. To convert a numeral string X into a number x, the function
NUMradix(X) is used. Viewing X as the string X[1] . . . X [m] with the most signifi-
cant numeral first, the function is defined as

NUMradix(X) = a
m

i=1
X[i] radixm - i = a

m - 1

i=0
X[m - i] radixi

Observe that 0 … NUMradix(X) 6 radixm and that 0 … X[i] 6 radix.

[x]s Converts an integer into a byte string; it is the string of s bytes that encodes the
number x, with 0 … x 6 28s. The equivalent notation is STR28s(x).

LEN(X) Length of the character string X.

NUMradix(X) Converts strings to numbers. The number that the numeral string X represents
in base radix, with the most significant character first. In other words, it is the
nonnegative integer less than radixLEN(X) whose most-significant-character-first
representation in base radix is X.

PRFK(X) A pseudorandom function that produces a 128-bit output with X as the input,
using encryption key K.

STRradix
m (x) Given a nonnegative integer x less than radixm, this function produces a repre-

sentation of x as a string of m characters in base radix, with the most significant
character first.

[i .. j] The set of integers between two integers i and j, including i and j.

X[i .. j] The substring of characters of a string X from X[i] to X[j], including X[i] and X[j].

REV(X) Given a bit string, X, the string that consists of the bits of X in reverse order.

(a) Notation

radix The base, or number of characters, in a given plaintext alphabet.

tweak Input parameter to the encryption and decryption functions whose confidentiality
is not protected by the mode.

tweakradix The base for tweak strings

minlen Minimum message length, in characters.

maxlen Maximum message length, in characters.

maxTlen Maximum tweak length

(b) Parameters

Table 7.3 Notation and Parameters Used in FPE Algorithms

7.8 / FORMAT-PRESERVING ENCRYPTION 237

For example, consider the string zaby in radix 26, which converts to the
numeral string 25 0 1 24. This converts to the number x = (25 * 263) + (1 * 261)
+ 2 4 = 4 3 9 4 5 0 . To go in the opposite direction and convert from a number
x 6 radixm to a numeral string X of length m, the function STRradixm (x) is used:

STRradix
m (x) = X[1]c X[m], where

X[i] = j x
radixm - i

kmod radix, i = 1, c, m
With the mapping of characters to numerals and the use of the NUM func-

tion, a plaintext character string can be mapped to a number and stored as an
unsigned integer. We would like to treat this unsigned integer as a bit string that
can be input to a bit-scrambling algorithm in FK. However, different platforms store
unsigned integers differently, some in little-endian and some in big-endian fashion.
So one more step is needed. By the definition of the STR function, STR2

8s(x) will
generate a bit string of length 8s, equivalently a byte string of length s, which is a
binary integer with the most significant bit first, regardless of how x is stored as an
unsigned integer. For convenience the following notation is used: [x]s = STR28s(x).
Thus, [NUMradix(X)]

s will convert the character string X into an unsigned integer
and then convert that to a byte string of length s bytes with the most significant
bit first.

Continuing, the preceding example should help clarify the issues involved.

Character string “zaby”

Numeral string X representation of
character string

25 0 1 24

Convert X to number
x = NUM26(X)

decimal: 439450
hex: 6B49A
binary: 1101011010010011010

x stored on big-endian byte order
machine as a 32-bit unsigned
integer

hex: 00 06 B4 9A
binary: 00000000000001101011010010011010

x stored on little-endian byte
order machine as a 32-bit unsigned
integer

hex: 9A B4 06 00
binary: 10011010101101000000011000000000

Convert x, regardless of endian
format, to a bit string of length
32 bits (4 bytes), expressed as [x]4

00000000000001101011010010011010

THE FUNCTION FK We can now define in general terms the function FK. The
core of FK is some type of randomizing function whose input and output are bit
strings. For convenience, the strings should be multiples of 8 bits, forming byte
strings. Define m to be u for even rounds and v for odd rounds; this specifies
the desired output character string length. Define b to be the number of bytes
needed to store the number representing a character string of m bytes. Then the

238 CHAPTER 7 / BLOCK CIPHER OPERATION

round, including FK, consists of the following general steps (A and B refer to Ai
and Bi for round i):

1. Q d [NUMradix(B)]b Converts numeral string X into byte string Q of
length b bytes.

2. Y d RAN[Q] A pseudorandom function PRNF that produces
a pseudorandom byte string Y as a function of
the bits of Q.

3. y d NUM2(Y) Converts Y into unsigned integer.

4. c d (NUMradix(A) + y) mod radixm Converts numeral string A into an integer and
adds to y, modulo radixm.

5. C d STRradixm (c) Converts c into a numeral string C of length m.
6. A d B;

B d C
Completes the round by placing the unchanged
value of B from the preceding round into A, and
placing C into B.

Steps 1 through 3 constitute the round function FK. Step 3 is presented with Y,
which is an unstructured bit string. Because different platforms may store unsigned
integers using different word lengths and endian conventions, it is necessary to per-
form NUM2(Y) to get an unsigned integer y. The stored bit sequence for y may or
may not be identical to the bit sequence for Y.

As mentioned, the pseudorandom function in step 2 need not be reversible. Its
purpose is to provide a randomized, scrambled bit string. For DES, this is achieved
by using fixed S-boxes, as described in Appendix S. Virtually all FPE schemes that
use the Feistel structure use AES as the basis for the scrambling function to achieve
stronger security.

RELATIONSHIP BETWEEN RADIX, MESSAGE LENGTH, AND BIT LENGTH Consider
a numeral string X of length len and base radix. If we convert this to a number
x = NUMradix(X), then the maximum value of x is radixlen - 1. The number of bits
needed to encode x is

bitlen = <LOG2(radixlen)= = <lenLOG2(radix)=
Observe that an increase in either radix or len increases bitlen. Often, we want

to limit the value of bitlen to some fixed upper limit, for example, 128 bits, which is
the size of the input to AES encryption. We also want the FPE to handle a variety of
radix values. The typical FPE, and all of those discussed subsequently, allow a given
range of radix values and then define a maximum character string length in order to
provide the algorithm with a fixed value of bitlen. Let the range of radix values be
from 2 to maxradix, and the maximum allowable character string value be maxlen.
Then the following relationship holds:

maxlen … :bitlen/LOG2(radix); , or equivalently
maxlen … :bitlen * LOGradix(2);

For example, for a radix of 10, maxlen … :0.3 * bitlen; ; for a radix of 26,
maxlen … :0.21 * bitlen; . The larger the radix, the smaller the maximum charac-
ter length for a given bit length.

7.8 / FORMAT-PRESERVING ENCRYPTION 239

NIST Methods for Format-Preserving Encryption

In 2013, NIST issued SP 800-38G: Recommendation for Block Cipher Modes of
Operation: Methods for Format-Preserving Encryption. This Recommendation
specifies three methods for format-preserving encryption, called FF1, FF2, and FF3.
The three methods all use the Feistel structure shown in Figure 7.12. They employ
somewhat different round functions FK, which are built using AES. Important dif-
ferences are the following:

■ FF1 supports the greatest range of lengths for the plaintext character string
and the tweak. To achieve this, the round function uses a cipher-block-chaining
(CBC) style of encryption, whereas FF2 and FF3 employ simple electronic
codebook (ECB) encryption.

■ FF2 uses a subkey generated from the encryption key and the tweak, whereas
FF1 and FF3 use the encryption key directly. The use of a subkey may help
protect the original key from side-channel analysis, which is an attack based
on information gained from the physical implementation of a cryptosystem,
rather than brute force or cryptanalysis. Examples of such attacks are attempts
to deduce key bits based on power consumption or execution time.

■ FF3 offers the lowest round count, eight, compared to ten for FF1 and FF2,
and is the least flexible in the tweaks that it supports.

ALGORITHM FF1 Algorithm FF1 was submitted to NIST as a proposed FPE mode
[BELL10a, BELL10b] with the name FFX[Radix]. FF1 uses a pseudorandom func-
tion PRFK(X) that produces a 128-bit output with inputs X that is a multiple of 128
bits and encryption key K (Figure 7.13). In essence, PRFK(X) use CBC encryption
(Figure 7.4) with X as the plaintext input, encryption key K, and an initial vector
(IV) of all zeros. The output is the last block of ciphertext produced. This is also

Figure 7.13 Algorithm PRF(X)

Prerequisites:

Approved, 128-bit block cipher, CIPH;
Key, K, for the block cipher;

Input:

Nonempty bit string, X, such that LEN(X) is a multiple of 128.
Output:
128-bit block, Y

Steps:

1. Let m = LEN(X)/128.
2. Partition X into m 128-bit blocks X1, c , Xm, so that X = X1 } c }Xm
3. Let Y0 = [0]16

4. For j from 1 to m:

4.i let Yj = CIPHK(Yj - 1⊕ Xj).
6. Return Ym.

240 CHAPTER 7 / BLOCK CIPHER OPERATION

equivalent to the message authentication code known as CBC-MAC, or CMAC,
described in Chapter 12.

The FF1 encryption algorithm is illustrated in Figure 7.14. The shaded lines
correspond to the function FK. The algorithm has 10 rounds and the following
parameters (Table 7.3b):

■ radix∈ [2 .. 216]
■ radixminlen Ú 100
■ minlen Ú 2
■ maxlen 6 232. For the maximum radix value of 216, the maximum bit length to

store the integer value of X is 16 * 232 bits; for the minimum radix value of 2,
the maximum bit length to store the integer value of X is 232 bits.

■ maxTlen 6 232

The inputs to the encryption algorithm are a character string X of length n
and a tweak T of length t. The tweak is optional in that it may be the empty string.

Prerequisites:

Approved, 128-bit block cipher, CIPH;
Key, K, for the block cipher;
Base, radix, for the character alphabet;
Range of supported message lengths, [minlen .. maxlen];
Maximum byte length for tweaks, maxTlen.

Inputs:

Character string, X, in base radix of length n such that n ∈ [minlen .. maxlen];
Tweak T, a byte string of byte length t, such that t ∈ [0 .. maxTlen].

Output:

Character string, Y, such that LEN(Y) = n.

Steps:

1. Let u = :n/2; ; v = n - u.
2. Let A = X[1 .. u]; B = X[u + 1 .. n].
3. Let b = < <v LOG2(radix)= /8= ; d = 4<b/4= + 4
4. Let P = [1]1 } [2]1 } [1]1 } [radix]3 } [10]1 } [u mod 256]1 } [n]4 } [t]4.
5. For i from 0 to 9:

i. Let Q = T } [0](-t - b - 1) mod 16 } [i]1 } [NUMradix(B)]b.
ii. Let R = PRFK(P}Q).

iii. Let S be the first d bytes of the following string of [d/16] 128-bit blocks:
R }CIPHK(R⊕ [1]16) }CIPHK(R⊕ [2]16) } c }CIPHK(R⊕ [<d/16= - 1]16).

iv. Let y = NUM2(S).
v. If i is even, let m = u; else, let m = v.

vi. Let c = (NUMradix(A) + y) mod radixm.
vii. Let C = STRradixm (c).

viii. Let A = B.
ix. Let B = C.

6. Return Y = A}B.

Figure 7.14 Algorithm FF1 (FFX[Radix])

7.8 / FORMAT-PRESERVING ENCRYPTION 241

The output is the encrypted character string Y of length n. What follows is a step-by-
step description of the algorithm.

1., 2. The input X is split into two substrings A and B. If n is even, A and B are
of equal length. Otherwise, B is one character longer than A.

3. The expression <v LOG2(radix)= equals the number of bits needed to
encode B, which is v characters long. Encoding B as a byte string, b is
the number of bytes in the encoding. The definition of d ensures that the
output of the Feistel round function is at least 4 bytes longer than this
encoding of B, which minimizes any bias in the modular reduction in
step 5.vi, as explained subsequently.

4. P is a 128-bit (16-byte) block that is a function of radix, u, n, and t. It
serves as the first block of plaintext input to the CBC encryption mode
used in 5.ii, and is intended to increase security.

5. The loop through the 10 rounds of encryption.

5.i The tweak, T, the substring, B, and the round number, i, are encoded
as a binary string, Q, which is one or more 128-bit blocks in length. To
understand this step, first note that the value NUMradix(B) produces a
numeral string that represents B in base radix. How this numeral string is
formatted and stored is outside the scope of the standard. Then, the value
[NUMradix(B)]

b produces the representation of the numerical value of B
as a binary number in a string of b bytes. We also have the length of T
is t bytes, and the round number is stored in a single byte. This yields a
length of (t + b + 1) bytes. This is padded out with z = (- t - b - 1)
mod 16 bytes. From the rules of modular arithmetic, we know that
(z + t + b + 1) mod 16 = 0. Thus the length of Q is one or more 128-
bit blocks.

5.ii The concatenation of P and Q is input to the pseudorandom func-
tion PRF to produce a 128-bit output R. This function is the pseudo-
random core of the Feistel round function. It scrambles the bits of Bi
(Figure 7.12).

5.iii This step either truncates or expands R to a byte string S of length d
bytes. That is, if d … 16 bytes, then R is the first d bytes of R. Otherwise
the 16-byte R is concatenated with successive encryptions of R XORed
with successive constants to produce the shortest string of 16-byte blocks
whose length is greater than or equal to d bytes.

5.iv This step begins the process of converting the results of the scrambling
of Bi into a form suitable for combining with Ai. In this step, the d-byte
string S is converted into a numeral string in base 2 that represents S.
That is, S is represented as a binary string y.

5.v This step determines the length m of the character string output that is
required to match the length of the B portion of the round output. For
even-numbered rounds, the length is u characters, and for odd- numbered
rounds it is v characters, as shown in Figure 7.12.

5.vi The numerical values of A and y are added modulo radixm. This trun-
cates the value of the sum to a value c that can be stored in m characters.

242 CHAPTER 7 / BLOCK CIPHER OPERATION

5.vii This step converts the c into the proper representation C as a string of m
characters.

5.viii, 5.ix These steps complete the round by placing the unchanged value of B
from the preceding round into A, and placing C into B.

6. After the final round, the result is returned as the concatenation of A and B.

It may be worthwhile to clarify the various uses of the NUM function in FF1.
NUM converts strings with a given radix into integers. In step 5.i, B is a character
string in base radix, so NUMradix(B) converts this into an integer, which is stored
as a byte string, suitable for encryption in step 5.ii. For step 5.iv, S is a byte string
output of an encryption function, which can be viewed a bit string, so NUM2(S)
converts this into an integer.

Finally, a brief explanation of the variable d is in order, which is best ex-
plained by example. Let radix = 26 and v = 30 characters. Then b = 18 bytes,
and d = 24 bytes. Step 5.ii produces an output R of 16 bytes. We desire a scram-
bled output of b bytes to match the input, and so R needs to be padded out. Rather
than padding with a constant value such as all zeros, step 5.iii pads out with random
bits. The result, in step 5.iv is a number greater than radixm of fully randomized
bits. The use of randomized padding avoids a potential security risk of using a fixed
padding.

ALGORITHM FF2 Algorithm FF2 was submitted to NIST as a proposed FPE
mode with the name VAES3 [VANC11]. The encryption algorithm is defined in
Figure 7.15. The shaded lines correspond to the function FK. The algorithm has the
following parameters:

■ radix∈ [2 .. 28]
■ tweakradix∈ [2 .. 28]
■ radixminlen Ú 100
■ minlen Ú 2
■ maxlen … 2:120/LOG2(radix); if radix is a power of 2. For the maximum radix

value of 28, maxlen … 30; for the minimum radix value of 2, maxlen … 240. In
both cases, the maximum bit length to store the integer value of X is 240 bits,
or 30 bytes.

■ maxlen … 2:98/LOG2(radix); if radix is a not a power of 2. For the maxi-
mum radix value of 255, maxlen … 24; for the minimum radix value of 3,
maxlen … 124.

■ maxTlen … :104/LOG2(tweakradix); . For the maximum tweakradix value of
28, maxTlen … 13.

For FF2, the plaintext character alphabet and that of the tweak may be different.
The first two steps of FF2 are the same as FF1, setting values for v, u, A, and B.

FF2 proceeds with the following steps:

3. P is a 128-bit (16-byte) block. If there is a tweak, then P is a function of
radix, t, n, and the 13-byte numerical value of the tweak. If there is no tweak
(t = 0), then P is a function of radix and n. P is used to form an encryption key
in step 4.

7.8 / FORMAT-PRESERVING ENCRYPTION 243

4. J is the encryption of P using the input key K.

5. The loop through the 10 rounds of encryption.

5.i B is converted into a 15-byte number, prepended by the round number to
form a 16-byte block Q.

5.ii Q is encrypted using the encryption key J to yield Y.

The remaining steps are the same as for FF1. The essential difference is in the
way in which all of the parameters are incorporated into the encryption that takes
place in the block FK. In both cases, the encryption is not simply an encryption of B
using key K. For FF1, B is combined with the tweak, the round number, t, n, u, and
radix to form a string of multiple 16-byte blocks. Then CBC encryption is used with
K to produce a 16-byte output. For FF2, all of the parameters besides B are com-
bined to form a 16-byte block, which is then encrypted with K to form the key value
J. J is then used as the key for the one-block encryption of B.

The structure of FF2 explains the maximum length restrictions. In step 3, P
incorporates the radix, tweak length, the numeral string length, and the tweak into
the calculation. As input to AES, P is restricted to 16 bytes. With a maximum radix
value of 28, the radix value can be stored in one byte (byte value 0 corresponds
to 256). The string length n and tweak length t each easily fits into one byte. This
leaves a restriction that the value of the tweak should be stored in at most 13 bytes,

Approved, 128-bit block cipher, CIPH;
Key, K, for the block cipher;
Base, tweakradix, for the tweak character alphabet;
Range of supported message lengths, [minlen .. maxlen];
Maximum supported tweak length, maxTlen.

Inputs:

Numeral string, X, in base radix, of length n such that n ∈ [minlen .. maxlen];
Tweak numeral string, T, in base tweakradix, of length t such that t ∈ [0 .. maxTlen].

Output:
Numeral string, Y, such that LEN(Y) = n.

Steps:

1. Let u = :n/2; ; v = n - u.
2. Let A = X[1 .. u]; B = X[u + 1 .. n].
3. If t 7 0, P = [radix]1 } [t]1 } [n]1 } [NUM tweakradix(T)]13; else P = [radix]1 } [0]1 } [n]1 } [0]13.
4. Let J = CIPHK(P).
5. For i from 0 to 9:

i. Let Q d [i]1 } [NUMradix(B)]15
ii. Let Y d CIPHJ(Q).

iii. Let y d NUM2(Y).
iv. If i is even, let m = u; else, let m = v.

v. Let c = (NUMradix(A) + y) mod radixm.
vi. Let C = STRradixm (c).

vii. Let A = B.
viii. Let B = C.

6. Return Y = A}B.

Figure 7.15 Algorithm FF2 (VAES3)

244 CHAPTER 7 / BLOCK CIPHER OPERATION

or 104 bits. The number of bits to store the tweak is LOG2(tweakradix
Tlen). This

leads to the restriction maxTlen Ú :104/LOG2(tweakradix); . Similarly step 5i
incorporates B and the round number into a 16-byte input to AES, leaving
15 bytes to encode B, or 120 bits, so that the length must be less than or equal to :120/LOG2(radix); . The parameter maxlen refers to the entire block, consisting of
partitions A and B, thus maxlen Ú 2:120/LOG2(radix); .

There is a further restriction on maxlen for a radix that is not a power of 2.
As explained in [VANC11], when the radix is not a power of 2, modular arithme-
tic causes the value (y mod radixm) to not have uniform distribution in the output
space, which can result in a cryptographic weakness.

ALGORITHM FF3 Algorithm FF3 was submitted to NIST as a proposed FPE mode
with the name BPS-BC [BRIE10]. The encryption algorithm is illustrated in
Figure 7.16. The shaded lines correspond to the function FK. The algorithm has the
following parameters:

■ radix∈ [2 .. 216]
■ radixminlen Ú 100
■ minlen Ú 2

Approved, 128-bit block cipher, CIPH;
Key, K, for the block cipher;
Base, radix, for the character alphabet such that radix ∈ [2..216];
Range of supported message lengths, [minlen .. maxlen], such that minlen Ú 2 and
maxlen … 2:logradix(296); .
Inputs:

Numeral string, X, in base radix of length n such that n ∈ [minlen .. maxlen];
Tweak bit string, T, such that LEN(T) = 64.

Output:
Numeral string, Y, such that LEN(Y) = n.

Steps:

1. Let u = <n/2= ; v = n - u.
2. Let A = X[1 .. u]; B = X[u + 1 .. n].
3. Let TL = T[0 .. 31] and TR = T[32 .. 63].
4. For i from 0 to 7:

i. If i is even, let m = u and W = TR, else let m = v and W = TL.
ii. Let P = REV([NUMradix(REV(B))]12) } [W⊕ REV([i]4]).

iii. Let Y = CIPHK(P).
iv. Let y = NUM2(REV(Y)).

v. Let c = (NUMradix(REV(A)) + y) mod radixm.
vi. Let C = REV(STRradixm (c)).

vii. Let A = B.
viii. Let B = C.

5. Return A}B.

Figure 7.16 Algorithm FF3 (BPS-BC)

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7.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 245

■ maxlen … 2:LOGradix(296); . For the maximum radix value of 216, maxlen … 12;
for the minimum radix value of 2, maxlen … 192. In both cases, the maximum
bit length to store the integer value of X is 192 bits, or 24 bytes.

■ Tweak length = 64 bits

FF3 proceeds with the following steps:

1., 2. The input X is split into two substrings A and B. If n is even, A and B are
of equal length. Otherwise, A is one character longer than B, in contrast
to FF1 and FF2, where B is one character longer than A.

3. The tweak is partitioned into a 32-bit left tweak TL and a 32-bit right
tweak TR.

4. The loop through the 8 rounds of encryption.

4.i As in FF1 and FF2, this step determines the length m of the character
string output that is required to match the length of the B portion of the
round output. The step also determines whether TL or TR will be used as
W in step 4ii.

4.ii The bits of B are reversed, then NUMradix(B) produces a 12-byte numeral
string in base radix; the results are again reversed. A 32-bit encoding of
the round number i is stored in a 4-byte unit, which is reversed and then
XORed with W. P is formed by concatenating these two results to form a
16-byte block.

4.iii P is encrypted using the encryption key K to yield Y.

4.iv This is similar to step 5.iv in FF1, except that Y is reversed before convert-
ing it into a numeral string in base 2.

4.v The numerical values of the reverse of A and y are added modulo radixm.
This truncates the value of the sum to a value c that can be stored in m
characters.

4.vi This step converts c to a numeral string C.

The remaining steps are the same as for FF1.

7.9 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

block cipher modes of
operation

cipher block chaining mode
(CBC)

cipher feedback mode
(CFB)

ciphertext stealing
counter mode (CTR)
electronic codebook mode

(ECB)
meet-in-the-middle attack
nonce

output feedback mode
(OFB)

Triple DES (3DES)
tweakable block cipher
XTS-AES mode

246 CHAPTER 7 / BLOCK CIPHER OPERATION

Review Questions

7.1 What is triple encryption?
7.2 What is a meet-in-the-middle attack?
7.3 How many keys are used in triple encryption?
7.4 List and briefly define the block cipher modes of operation.
7.5 Why do some block cipher modes of operation only use encryption while others use

both encryption and decryption?

Problems

7.1 You want to build a hardware device to do block encryption in the cipher block chain-
ing (CBC) mode using an algorithm stronger than DES. 3DES is a good candidate.
Figure 7.17 shows two possibilities, both of which follow from the definition of CBC.
Which of the two would you choose:
a. For security?
b. For performance?

7.2 Can you suggest a security improvement to either option in Figure 7.17, using only
three DES chips and some number of XOR functions? Assume you are still limited to
two keys.

Figure 7.17 Use of Triple DES in CBC Mode

Pn

K1 E

An-1

An

K2 D

K1 E

Bn-1

Bn
Cn-1

Cn

(b) Three-loop CBC

Pn

K1, K2 EDE

Cn-1

Cn

(a) One-loop CBC

7.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 247

7.3 The Merkle–Hellman attack on 3DES begins by assuming a value of A = 0
(Figure 7.1b). Then, for each of the 256 possible values of K1, the plaintext P that
produces A = 0 is determined. Describe the rest of the algorithm.

7.4 With the ECB mode, if there is an error in a block of the transmitted ciphertext, only
the corresponding plaintext block is affected. However, in the CBC mode, this error
propagates. For example, an error in the transmitted C1 (Figure 7.4) obviously cor-
rupts P1 and P2.
a. Are any blocks beyond P2 affected?
b. Suppose that there is a bit error in the source version of P1. Through how many

ciphertext blocks is this error propagated? What is the effect at the receiver?
7.5 Is it possible to perform encryption operations in parallel on multiple blocks of plain-

text in CBC mode? How about decryption?
7.6 CBC-Pad is a block cipher mode of operation used in the RC5 block cipher, but it

could be used in any block cipher. CBC-Pad handles plaintext of any length. The
ciphertext is longer then the plaintext by at most the size of a single block. Padding is
used to assure that the plaintext input is a multiple of the block length. It is assumed
that the original plaintext is an integer number of bytes. This plaintext is padded at
the end by from 1 to bb bytes, where bb equals the block size in bytes. The pad bytes
are all the same and set to a byte that represents the number of bytes of padding. For
example, if there are 8 bytes of padding, each byte has the bit pattern 00001000. Why
not allow zero bytes of padding? That is, if the original plaintext is an integer multiple
of the block size, why not refrain from padding?

7.7 For the ECB, CBC, and CFB modes, the plaintext must be a sequence of one or more
complete data blocks (or, for CFB mode, data segments). In other words, for these
three modes, the total number of bits in the plaintext must be a positive multiple of
the block (or segment) size. One common method of padding, if needed, consists of a
1 bit followed by as few zero bits, possibly none, as are necessary to complete the final
block. It is considered good practice for the sender to pad every message, including
messages in which the final message block is already complete. What is the motiva-
tion for including a padding block when padding is not needed?

7.8 If a bit error occurs in the transmission of a ciphertext character in 8-bit CFB mode,
how far does the error propagate?

7.9 In discussing OFB, it was mentioned that if it was known that two different messages
had an identical block of plaintext in the identical position, it is possible to recover
the corresponding Oi block. Show the calculation.

7.10 In discussing the CTR mode, it was mentioned that if any plaintext block that is
encrypted using a given counter value is known, then the output of the encryption
function can be determined easily from the associated ciphertext block. Show the
calculation.

7.11 Padding may not always be appropriate. For example, one might wish to store the
encrypted data in the same memory buffer that originally contained the plaintext. In that
case, the ciphertext must be the same length as the original plaintext. We saw the use
of ciphertext stealing in the case of XTS-AES to deal with partial blocks. Figure 7.18a
shows the use of ciphertext stealing to modify CBC mode, called CBC-CTS.
a. Explain how it works.
b. Describe how to decrypt Cn - 1 and Cn.

7.12 Figure 7.18b shows an alternative to CBC-CTS for producing ciphertext of equal
length to the plaintext when the plaintext is not an integer multiple of the block size.
a. Explain the algorithm.
b. Explain why CBC-CTS is preferable to this approach illustrated in Figure 7.18b.

7.13 Draw a figure similar to those of Figure 7.8 for XTS-AES mode.
7.14 Work out the following problems from first principles without converting to binary

and counting the bits. Then compare with the formulae presented for encoding a

248 CHAPTER 7 / BLOCK CIPHER OPERATION

character string into an integer, and vice-versa, in the specified radix. (Hint: Consider
the next-lower and next-higher power of two for each integer.)
a. How many bits are exactly required to encode the following integers? (The num-

ber shown as an integer’s subscript refers to the radix of that integer.)
i. 2 0 4 71 0

ii. 2 0 4 8 1 0
iii. 3 2 7 6 71 0
iv. 3 2 7 6 8 1 0
v. 3 2 7 6 71 6

vi. 3 2 7 6 8 1 6
vii. 5 3 7 F1 6

viii. 2 9 4 3 11 0
b. Exactly how many bytes are required to represent the numbers in (a) above?

7.15 a. In radix-26, write down the numeral string X for each of the following character
strings, followed by the number of “digits” (i.e., the length of the numeral string)
in each case.
i. “hex”

ii. “cipher”
iii. “not”
iv. “symbol”

Figure 7.18 Block Cipher Modes for Plaintext not a Multiple of Block Size

IV P1

C1

K K K K

+

PN-2

CN-2

CN-3 +

Encrypt Encrypt Encrypt Encrypt

Encrypt Encrypt

(a) Ciphertext stealing mode

(b) Alternative method

Encrypt Encrypt

+

CN X

PN-1

+

CN-1

PN 00…0

IV
P1

(bb bits)

C1
(bb bits)

K K K K

+

PN-2
(bb bits)

CN-2
(bb bits)

CN-3 +

select
leftmost

j bits

PN-1
(bb bits)

CN-1
(bb bits)

+

PN
(j bits)

CN
(j bits)

+

7.9 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 249

b. For each case of problem (a), determine the number x = NUM26(X)
c. Determine the byte form [x] for each number x computed in problem (b).
d. What is the smallest power of the radix (26) that is greater than each of the nu-

merical strings determined in (b)?
e. Is it related to the length of the numeral string in each case, in problem (a)? If so,

what is this relationship?
7.16 Refer to algorithms FF1 and FF2.

a. For step 1, for each algorithm, u d :n/2; and v d <n - u= . Show that for any
three integers x, y, and n:

if x = :n/2; and y = <n - x= , then:
i. Either x = n/2, or x = (n - 1)/2.

ii. Either y = n/2, or y = (n + 1)/2.
iii. x … y. (Under what condition is x = y?)

b. What is the significance of result in the previous sub-problem (iii), in terms of the
lengths u and v of the left and right half-strings, respectively?

7.17 In step 3 of Algorithm FF1, what do b and d represent? What is the unit of measure-
ment (bits, bytes, digits, characters) of each of these quantities?

7.18 In the inputs to algorithms FF1, FF2, and FF3, why are the specified radix ranges
important? For example, why should radix∈ [0..28] for Algorithm FF2, or
radix∈ [2..216] in the case of Algorithm FF3?

Programming Problems

7.19 Create software that can encrypt and decrypt in cipher block chaining mode using one
of the following ciphers: affine modulo 256, Hill modulo 256, S-DES, DES.

Test data for S-DES using a binary initialization vector of 1010 1010. A binary plain-
text of 0000 0001 0010 0011 encrypted with a binary key of 01111 11101 should give
a binary plaintext of 1111 0100 0000 1011. Decryption should work correspondingly.

7.20 Create software that can encrypt and decrypt in 4-bit cipher feedback mode using one
of the following ciphers: additive modulo 256, affine modulo 256, S-DES;

or
8-bit cipher feedback mode using one of the following ciphers: 2 * 2 Hill modulo 256.

Test data for S-DES using a binary initialization vector of 1010 1011. A binary plain-
text of 0001 0010 0011 0100 encrypted with a binary key of 01111 11101 should give
a binary plaintext of 1110 1100 1111 1010. Decryption should work correspondingly.

7.21 Create software that can encrypt and decrypt in counter mode using one of the follow-
ing ciphers: affine modulo 256, Hill modulo 256, S-DES.

Test data for S-DES using a counter starting at 0000 0000. A binary plaintext of 0000
0001 0000 0010 0000 0100 encrypted with a binary key of 01111 11101 should give
a binary plaintext of 0011 1000 0100 1111 0011 0010. Decryption should work cor-
respondingly.

7.22 Implement a differential cryptanalysis attack on 3-round S-DES.

250250

8.1 Principles of Pseudorandom Number Generation

The Use of Random Numbers
TRNGs, PRNGs, and PRFs
PRNG Requirements
Algorithm Design

8.2 Pseudorandom Number Generators

Linear Congruential Generators
Blum Blum Shub Generator

8.3 Pseudorandom Number Generation Using a Block Cipher

PRNG Using Block Cipher Modes of Operation
ANSI X9.17 PRNG
NIST CTR_DRBG

8.4 Stream Ciphers

8.5 RC4

Initialization of S
Stream Generation
Strength of RC4

8.6 True Random Number Generators

Entropy Sources
Comparison of PRNGs and TRNGs
Conditioning
Health Testing
Intel Digital Random Number Generator

8.7 Key Terms, Review Questions, and Problems

CHAPTER

Random Bit Generation
and Stream Ciphers

RANDOM BIT GENERATION AND STREAM CIPHERS 251

An important cryptographic function is the generation of random bit streams. Random
bits streams are used in a wide variety of contexts, including key generation and
encryption. In essence, there are two fundamentally different strategies for generating
random bits or random numbers. One strategy, which until recently dominated in cryp-
tographic applications, computes bits deterministically using an algorithm. This class of
random bit generators is known as pseudorandom number generators (PRNGs) or
deterministic random bit generators (DRBGs). The other strategy is to produce bits
non-deterministically using some physical source that produces some sort of random
output. This latter class of random bit generators is known as true random number
generators (TRNGs) or non-deterministic random bit generators (NRBGs).

The chapter begins with an analysis of the basic principles of PRNGs. Next, we
look at some common PRNGs, including PRNGs based on the use of a symmetric
block cipher. The chapter then moves on to the topic of symmetric stream ciphers,
which are based on the use of a PRNG. The chapter next examines the most important
stream cipher, RC4.

The remainder of the chapter is devoted to TRNGs. We look first at the basic
principles and structure of TRNGs, and then examine a specific product, the Intel
Digital Random Number Generator.

Throughout this chapter, reference is made to four important NIST documents:

■ SP 800-90A (Recommendation for Random Number Generation Using
Deterministic Random Bit Generators, January 2012) covers DRNGs.

■ SP 800-90B (Recommendation for the Entropy Sources Used for Random Bit
Generation, August 2012) covers criteria for entropy sources (ES), the devices
from which we get unpredictable randomness and NRNGs.

■ SP 800-90C (Recommendation for Random Bit Generator (RBG)
Constructions, August 2012) discusses how to combine the entropy sources in
90B with the DRNG’s from 90A to provide large quantities of unpredictable
bits for cryptographic applications.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Explain the concepts of randomness and unpredictability with respect to
random numbers.

◆ Understand the differences among true random number generators,
pseudorandom number generators, and pseudorandom functions.

◆ Present an overview of requirements for pseudorandom number generators.

◆ Explain how a block cipher can be used to construct a pseudorandom
number generator.

◆ Present an overview of stream ciphers and RC4.

◆ Explain the significance of skew.

252 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

■ SP 800-22 (A Statistical Test Suite for Random and Pseudorandom Number
Generators for Cryptographic Applications, April 2010) discusses the selection
and testing of NRBGs and DRBGs.

These specifications have heavily influenced the implementation of random bit
generators in industry both in the U.S. and worldwide.

8.1 PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION

Random numbers play an important role in the use of encryption for various net-
work security applications. In this section, we provide a brief overview of the use
of random numbers in cryptography and network security and then focus on the
principles of pseudorandom number generation.

The Use of Random Numbers

A number of network security algorithms and protocols based on cryptography
make use of random binary numbers. For example,

■ Key distribution and reciprocal (mutual) authentication schemes, such as
those discussed in Chapters 14 and 15. In such schemes, two communicating
parties cooperate by exchanging messages to distribute keys and/or authen-
ticate each other. In many cases, nonces are used for handshaking to prevent
replay attacks. The use of random numbers for the nonces frustrates an oppo-
nent’s efforts to determine or guess the nonce, in order to repeat an obsolete
transaction.

■ Session key generation. We will see a number of protocols in this book where a
secret key for symmetric encryption is generated for use for a particular trans-
action (or session) and is valid for a short period of time. This key is generally
called a session key.

■ Generation of keys for the RSA public-key encryption algorithm (described
in Chapter 9).

■ Generation of a bit stream for symmetric stream encryption (described in this
chapter).

These applications give rise to two distinct and not necessarily compatible
requirements for a sequence of random numbers: randomness and unpredictability.

RANDOMNESS Traditionally, the concern in the generation of a sequence of alleg-
edly random numbers has been that the sequence of numbers be random in some
well-defined statistical sense. The following two criteria are used to validate that a
sequence of numbers is random:

■ Uniform distribution: The distribution of bits in the sequence should be
uniform; that is, the frequency of occurrence of ones and zeros should be
approximately equal.

■ Independence: No one subsequence in the sequence can be inferred from the
others.

8.1 / PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION 253

Although there are well-defined tests for determining that a sequence of bits
matches a particular distribution, such as the uniform distribution, there is no such
test to “prove” independence. Rather, a number of tests can be applied to demon-
strate if a sequence does not exhibit independence. The general strategy is to apply
a number of such tests until the confidence that independence exists is sufficiently
strong. That is, if each of a number of tests fails to show that a sequence of bits is
not independent, then we can have a high level of confidence that the sequence is in
fact independent.

In the context of our discussion, the use of a sequence of numbers that appear
statistically random often occurs in the design of algorithms related to cryptography.
For example, a fundamental requirement of the RSA public-key encryption scheme
discussed in Chapter 9 is the ability to generate prime numbers. In general, it is
difficult to determine if a given large number N is prime. A brute-force approach
would be to divide N by every odd integer less than 2N. If N is on the order, say,
of 10150, which is a not uncommon occurrence in public-key cryptography, such a
brute-force approach is beyond the reach of human analysts and their computers.
However, a number of effective algorithms exist that test the primality of a num-
ber by using a sequence of randomly chosen integers as input to relatively simple
computations. If the sequence is sufficiently long (but far, far less than 210150), the
primality of a number can be determined with near certainty. This type of approach,
known as randomization, crops up frequently in the design of algorithms. In es-
sence, if a problem is too hard or time-consuming to solve exactly, a simpler, shorter
approach based on randomization is used to provide an answer with any desired
level of confidence.

UNPREDICTABILITY In applications such as reciprocal authentication, session key
generation, and stream ciphers, the requirement is not just that the sequence of
numbers be statistically random but that the successive members of the sequence
are unpredictable. With “true” random sequences, each number is statistically inde-
pendent of other numbers in the sequence and therefore unpredictable. Although
true random numbers are used in some applications, they have their limitations,
such as inefficiency, as is discussed shortly. Thus, it is more common to imple-
ment algorithms that generate sequences of numbers that appear to be random. In
this latter case, care must be taken that an opponent not be able to predict future
elements of the sequence on the basis of earlier elements.

TRNGs, PRNGs, and PRFs

Cryptographic applications typically make use of algorithmic techniques for ran-
dom number generation. These algorithms are deterministic and therefore produce
sequences of numbers that are not statistically random. However, if the algorithm is
good, the resulting sequences will pass many tests of randomness. Such numbers are
referred to as pseudorandom numbers.

You may be somewhat uneasy about the concept of using numbers generated
by a deterministic algorithm as if they were random numbers. Despite what might be
called philosophical objections to such a practice, it generally works. That is, under
most circumstances, pseudorandom numbers will perform as well as if they were
random for a given use. The phrase “as well as” is unfortunately subjective, but the

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254 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

use of pseudorandom numbers is widely accepted. The same principle applies in
statistical applications, in which a statistician takes a sample of a population and
assumes that the results will be approximately the same as if the whole population
were measured.

Figure 8.1 contrasts a true random number generator (TRNG) with two forms
of pseudorandom number generators. A TRNG takes as input a source that is
effectively random; the source is often referred to as an entropy source. We discuss
such sources in Section 8.6. In essence, the entropy source is drawn from the physi-
cal environment of the computer and could include things such as keystroke timing
patterns, disk electrical activity, mouse movements, and instantaneous values of the
system clock. The source, or combination of sources, serve as input to an algorithm
that produces random binary output. The TRNG may simply involve conversion of
an analog source to a binary output. The TRNG may involve additional processing
to overcome any bias in the source; this is discussed in Section 8.6.

In contrast, a PRNG takes as input a fixed value, called the seed, and produces
a sequence of output bits using a deterministic algorithm. Quite often, the seed is
generated by a TRNG. Typically, as shown, there is some feedback path by which
some of the results of the algorithm are fed back as input as additional output bits
are produced. The important thing to note is that the output bit stream is deter-
mined solely by the input value or values, so that an adversary who knows the algo-
rithm and the seed can reproduce the entire bit stream.

Figure 8.1 shows two different forms of PRNGs, based on application.

■ Pseudorandom number generator: An algorithm that is used to produce an
open-ended sequence of bits is referred to as a PRNG. A common application
for an open-ended sequence of bits is as input to a symmetric stream cipher,
as discussed in Section 8.4. Also, see Figure 4.1a.

Figure 8.1 Random and Pseudorandom Number Generators

Conversion
to binary

Source of
true

randomness

Random
bit stream

(a) TRNG

TRNG = true random number generator
PRNG = pseudorandom number generator
PRF = pseudorandom function

Deterministic
algorithm

Seed

Pseudorandom
bit stream

(b) PRNG

Deterministic
algorithm

Seed

Pseudorandom
value

(c) PRF

Context-
specific
values

8.1 / PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION 255

■ Pseudorandom function (PRF): A PRF is used to produce a pseudorandom
string of bits of some fixed length. Examples are symmetric encryption keys
and nonces. Typically, the PRF takes as input a seed plus some context specific
values, such as a user ID or an application ID. A number of examples of PRFs
will be seen throughout this book, notably in Chapters 17 and 18.

Other than the number of bits produced, there is no difference between a
PRNG and a PRF. The same algorithms can be used in both applications. Both
require a seed and both must exhibit randomness and unpredictability. Further,
a PRNG application may also employ context-specific input. In what follows, we
make no distinction between these two applications.

PRNG Requirements

When a PRNG or PRF is used for a cryptographic application, then the basic
requirement is that an adversary who does not know the seed is unable to determine
the pseudorandom string. For example, if the pseudorandom bit stream is used in
a stream cipher, then knowledge of the pseudorandom bit stream would enable the
adversary to recover the plaintext from the ciphertext. Similarly, we wish to pro-
tect the output value of a PRF. In this latter case, consider the following scenario.
A 128-bit seed, together with some context-specific values, are used to generate a
128-bit secret key that is subsequently used for symmetric encryption. Under nor-
mal circumstances, a 128-bit key is safe from a brute-force attack. However, if the
PRF does not generate effectively random 128-bit output values, it may be possible
for an adversary to narrow the possibilities and successfully use a brute force attack.

This general requirement for secrecy of the output of a PRNG or PRF leads
to specific requirements in the areas of randomness, unpredictability, and the char-
acteristics of the seed. We now look at these in turn.

RANDOMNESS In terms of randomness, the requirement for a PRNG is that the gen-
erated bit stream appear random even though it is deterministic. There is no single
test that can determine if a PRNG generates numbers that have the characteristic
of randomness. The best that can be done is to apply a sequence of tests to the
PRNG. If the PRNG exhibits randomness on the basis of multiple tests, then it can
be assumed to satisfy the randomness requirement. NIST SP 800-22 specifies that
the tests should seek to establish the following three characteristics.

■ Uniformity: At any point in the generation of a sequence of random or pseu-
dorandom bits, the occurrence of a zero or one is equally likely, that is, the
probability of each is exactly 1/2. The expected number of zeros (or ones) is
n/2, where n = the sequence length.

■ Scalability: Any test applicable to a sequence can also be applied to subse-
quences extracted at random. If a sequence is random, then any such extracted
subsequence should also be random. Hence, any extracted subsequence should
pass any test for randomness.

■ Consistency: The behavior of a generator must be consistent across starting
values (seeds). It is inadequate to test a PRNG based on the output from
a single seed or a TRNG on the basis of an output produced from a single
physical output.

256 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

SP 800-22 lists 15 separate tests of randomness. An understanding of these
tests requires a basic knowledge of statistical analysis, so we don’t attempt a techni-
cal description here. Instead, to give some flavor for the tests, we list three of the
tests and the purpose of each test, as follows.

■ Frequency test: This is the most basic test and must be included in any test
suite. The purpose of this test is to determine whether the number of ones and
zeros in a sequence is approximately the same as would be expected for a truly
random sequence.

■ Runs test: The focus of this test is the total number of runs in the sequence,
where a run is an uninterrupted sequence of identical bits bounded before
and after with a bit of the opposite value. The purpose of the runs test is to
determine whether the number of runs of ones and zeros of various lengths is
as expected for a random sequence.

■ Maurer’s universal statistical test: The focus of this test is the number of
bits between matching patterns (a measure that is related to the length of a
compressed sequence). The purpose of the test is to detect whether or not
the sequence can be significantly compressed without loss of information.
A significantly compressible sequence is considered to be non-random.

UNPREDICTABILITY A stream of pseudorandom numbers should exhibit two forms
of unpredictability:

■ Forward unpredictability: If the seed is unknown, the next output bit in the
sequence should be unpredictable in spite of any knowledge of previous bits
in the sequence.

■ Backward unpredictability: It should also not be feasible to determine the
seed from knowledge of any generated values. No correlation between a seed
and any value generated from that seed should be evident; each element of the
sequence should appear to be the outcome of an independent random event
whose probability is 1/2.

The same set of tests for randomness also provide a test of unpredictability. If
the generated bit stream appears random, then it is not possible to predict some bit
or bit sequence from knowledge of any previous bits. Similarly, if the bit sequence
appears random, then there is no feasible way to deduce the seed based on the bit
sequence. That is, a random sequence will have no correlation with a fixed value
(the seed).

SEED REQUIREMENTS For cryptographic applications, the seed that serves as input to
the PRNG must be secure. Because the PRNG is a deterministic algorithm, if the
adversary can deduce the seed, then the output can also be determined. Therefore,
the seed must be unpredictable. In fact, the seed itself must be a random or pseudo-
random number.

Typically, the seed is generated by a TRNG, as shown in Figure 8.2. This is
the scheme recommended by SP 800-90A. The reader may wonder, if a TRNG is
available, why it is necessary to use a PRNG. If the application is a stream cipher,
then a TRNG is not practical. The sender would need to generate a keystream of

8.1 / PRINCIPLES OF PSEUDORANDOM NUMBER GENERATION 257

bits as long as the plaintext and then transmit the keystream and the ciphertext
securely to the receiver. If a PRNG is used, the sender need only find a way to
deliver the stream cipher key, which is typically 54 or 128 bits, to the receiver in a
secure fashion.

Even in the case of a PRF application, in which only a limited number of bits
is generated, it is generally desirable to use a TRNG to provide the seed to the
PRF and use the PRF output rather than use the TRNG directly. As is explained in
Section 8.6, a TRNG may produce a binary string with some bias. The PRF would
have the effect of conditioning the output of the TRNG so as to eliminate that bias.

Finally, the mechanism used to generate true random numbers may not be
able to generate bits at a rate sufficient to keep up with the application requiring
the random bits.

Algorithm Design

Cryptographic PRNGs have been the subject of much research over the years,
and a wide variety of algorithms have been developed. These fall roughly into two
categories.

■ Purpose-built algorithms: These are algorithms designed specifically and
solely for the purpose of generating pseudorandom bit streams. Some of these
algorithms are used for a variety of PRNG applications; several of these are
described in the next section. Others are designed specifically for use in a
stream cipher. The most important example of the latter is RC4, described in
Section 8.5.

■ Algorithms based on existing cryptographic algorithms: Cryptographic
algorithms have the effect of randomizing input data. Indeed, this is a require-
ment of such algorithms. For example, if a symmetric block cipher produced
ciphertext that had certain regular patterns in it, it would aid in the process of

Figure 8.2 Generation of Seed Input to PRNG

Entropy
source

Pseudorandom
number generator

(PRNG)

Seed

Pseudorandom
bit stream

True random
number generator

(TRNG)

258 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

cryptanalysis. Thus, cryptographic algorithms can serve as the core of PRNGs.
Three broad categories of cryptographic algorithms are commonly used to
create PRNGs:

–Symmetric block ciphers: This approach is discussed in Section 8.3.

–Asymmetric ciphers: The number theoretic concepts used for an asymmet-
ric cipher can also be adapted for a PRNG; this approach is examined in
Chapter 10.
–Hash functions and message authentication codes: This approach is exam-
ined in Chapter 12.

Any of these approaches can yield a cryptographically strong PRNG.
A purpose-built algorithm may be provided by an operating system for general use.
For applications that already use certain cryptographic algorithms for encryption or
authentication, it makes sense to reuse the same code for the PRNG. Thus, all of
these approaches are in common use.

8.2 PSEUDORANDOM NUMBER GENERATORS

In this section, we look at two types of algorithms for PRNGs.

Linear Congruential Generators

A widely used technique for pseudorandom number generation is an algorithm first
proposed by Lehmer [LEHM51], which is known as the linear congruential method.
The algorithm is parameterized with four numbers, as follows:

m the modulus m 7 0
a the multiplier 0 6 a 6 m
c the increment 0 … c 6 m
X0 the starting value, or seed 0 … X0 6 m

The sequence of random numbers {Xn} is obtained via the following iterative
equation:

Xn + 1 = (aXn + c) mod m

If m, a, c, and X0 are integers, then this technique will produce a sequence of inte-
gers with each integer in the range 0 … Xn 6 m.

The selection of values for a, c, and m is critical in developing a good ran-
dom number generator. For example, consider a = c = 1. The sequence produced
is obviously not satisfactory. Now consider the values a = 7, c = 0, m = 32, and
X0 = 1. This generates the sequence {7, 17, 23, 1, 7, etc.}, which is also clearly
unsatisfactory. Of the 32 possible values, only four are used; thus, the sequence
is said to have a period of 4. If, instead, we change the value of a to 5, then the
sequence is {5, 25, 29, 17, 21, 9, 13, 1, 5, etc. }, which increases the period to 8.

We would like m to be very large, so that there is the potential for producing
a long series of distinct random numbers. A common criterion is that m be nearly

8.2 / PSEUDORANDOM NUMBER GENERATORS 259

equal to the maximum representable nonnegative integer for a given computer.
Thus, a value of m near to or equal to 231 is typically chosen.

[PARK88] proposes three tests to be used in evaluating a random number
generator:

T1: The function should be a full-period generating function. That is, the function
should generate all the numbers from 0 through m - 1 before repeating.

T2: The generated sequence should appear random.
T3: The function should implement efficiently with 32-bit arithmetic.

With appropriate values of a, c, and m, these three tests can be passed. With
respect to T1, it can be shown that if m is prime and c = 0, then for certain values
of a the period of the generating function is m - 1, with only the value 0 missing.
For 32-bit arithmetic, a convenient prime value of m is 231 - 1. Thus, the generating
function becomes

Xn + 1 = (aXn) mod (231 - 1)

Of the more than 2 billion possible choices for a, only a handful of multipliers
pass all three tests. One such value is a = 75 = 16807, which was originally selected
for use in the IBM 360 family of computers [LEWI69]. This generator is widely
used and has been subjected to a more thorough testing than any other PRNG. It is
frequently recommended for statistical and simulation work (e.g., [JAIN91]).

The strength of the linear congruential algorithm is that if the multiplier and
modulus are properly chosen, the resulting sequence of numbers will be statistically
indistinguishable from a sequence drawn at random (but without replacement) from
the set 1, 2, c , m - 1. But there is nothing random at all about the algorithm, apart
from the choice of the initial value X0. Once that value is chosen, the remaining num-
bers in the sequence follow deterministically. This has implications for cryptanalysis.

If an opponent knows that the linear congruential algorithm is being used and
if the parameters are known (e.g., a = 75, c = 0, m = 231 - 1), then once a single
number is discovered, all subsequent numbers are known. Even if the opponent
knows only that a linear congruential algorithm is being used, knowledge of a small
part of the sequence is sufficient to determine the parameters of the algorithm.
Suppose that the opponent is able to determine values for X0, X1, X2, and X3. Then

X1 = (aX0 + c) mod m
X2 = (aX1 + c) mod m
X3 = (aX2 + c) mod m

These equations can be solved for a, c, and m.
Thus, although it is nice to be able to use a good PRNG, it is desirable to make

the actual sequence used nonreproducible, so that knowledge of part of the se-
quence on the part of an opponent is insufficient to determine future elements of the
sequence. This goal can be achieved in a number of ways. For example, [BRIG79]
suggests using an internal system clock to modify the random number stream. One
way to use the clock would be to restart the sequence after every N numbers using
the current clock value (mod m) as the new seed. Another way would be simply to
add the current clock value to each random number (mod m).

260 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

Blum Blum Shub Generator

A popular approach to generating secure pseudorandom numbers is known as
the Blum Blum Shub (BBS) generator (see Figure 8.3), named for its developers
[BLUM86]. It has perhaps the strongest public proof of its cryptographic strength
of any purpose-built algorithm. The procedure is as follows. First, choose two large
prime numbers, p and q, that both have a remainder of 3 when divided by 4. That is,

p K q K 3(mod 4)

This notation, explained more fully in Chapter 4, simply means that (p mod 4) =
(q mod 4) = 3. For example, the prime numbers 7 and 11 satisfy 7 K 11 K 3(mod 4).
Let n = p * q. Next, choose a random number s, such that s is relatively prime to n;
this is equivalent to saying that neither p nor q is a factor of s. Then the BBS genera-
tor produces a sequence of bits Bi according to the following algorithm:

X0 = s2 mod n
for i = 1 to ∞

Xi = (Xi−1)2 mod n
Bi = Xi mod 2

Thus, the least significant bit is taken at each iteration. Table 8.1 shows an example
of BBS operation. Here, n = 192649 = 383 * 503, and the seed s = 101355.

The BBS is referred to as a cryptographically secure pseudorandom bit
generator (CSPRBG). A CSPRBG is defined as one that passes the next-bit test,
which, in turn, is defined as follows [MENE97]: A pseudorandom bit generator is
said to pass the next-bit test if there is not a polynomial-time algorithm1 that, on
input of the first k bits of an output sequence, can predict the (k + 1)st bit with
probability significantly greater than 1/2. In other words, given the first k bits of the

1A polynomial-time algorithm of order k is one whose running time is bounded by a polynomial of order k.

Figure 8.3 Blum Blum Shub Block Diagram

Generate
x2 mod n

Select least
significant bit

Initialize
with seed s

[0, 1]

8.3 / PSEUDORANDOM NUMBER GENERATION USING A BLOCK CIPHER 261

sequence, there is not a practical algorithm that can even allow you to state that the
next bit will be 1 (or 0) with probability greater than 1/2. For all practical purposes,
the sequence is unpredictable. The security of BBS is based on the difficulty of
factoring n. That is, given n, we need to determine its two prime factors p and q.

8.3 PSEUDORANDOM NUMBER GENERATION USING
A BLOCK CIPHER

A popular approach to PRNG construction is to use a symmetric block cipher as
the heart of the PRNG mechanism. For any block of plaintext, a symmetric block
cipher produces an output block that is apparently random. That is, there are no
patterns or regularities in the ciphertext that provide information that can be used
to deduce the plaintext. Thus, a symmetric block cipher is a good candidate for
building a pseudorandom number generator.

If an established, standardized block cipher is used, such as DES or AES, then
the security characteristics of the PRNG can be established. Further, many applica-
tions already make use of DES or AES, so the inclusion of the block cipher as part
of the PRNG algorithm is straightforward.

PRNG Using Block Cipher Modes of Operation

Two approaches that use a block cipher to build a PNRG have gained widespread
acceptance: the CTR mode and the OFB mode. The CTR mode is recommended in
NIST SP 800-90A, in the ANSI standard X9.82 (Random Number Generation), and
in RFC 4086 (Randomness Requirements for Security, June 2005). The OFB mode is
recommended in X9.82 and RFC 4086.

Figure 8.4 illustrates the two methods. In each case, the seed consists of two
parts: the encryption key value and a value V that will be updated after each block
of pseudorandom numbers is generated. Thus, for AES-128, the seed consists of a
128-bit key and a 128-bit V value. In the CTR case, the value of V is incremented

Table 8.1 Example Operation of BBS Generator

i Xi Bi
0 20749
1 143135 1
2 177671 1
3 97048 0
4 89992 0
5 174051 1
6 80649 1
7 45663 1
8 69442 0
9 186894 0
10 177046 0

i Xi Bi
11 137922 0
12 123175 1
13 8630 0
14 114386 0
15 14863 1
16 133015 1
17 106065 1
18 45870 0
19 137171 1
20 48060 0

262 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

by 1 after each encryption. In the case of OFB, the value of V is updated to equal the
value of the preceding PRNG block. In both cases, pseudorandom bits are produced
one block at a time (e.g., for AES, PRNG bits are generated 128 bits at a time).

The CTR algorithm for PRNG, called CTR_DRBG, can be summarized
as follows.

while (len (temp) < requested_number_of_bits) do
V = (V + 1) mod 2128

output_block = E(Key, V)
temp = temp || output_block

The OFB algorithm can be summarized as follows.

while (len (temp) < requested_number_of_bits) do
V = E(Key, V)
temp = temp || V

To get some idea of the performance of these two PRNGs, consider the fol-
lowing short experiment. A random bit sequence of 256 bits was obtained from
random.org, which uses three radios tuned between stations to pick up atmospheric
noise. These 256 bits form the seed, allocated as

Key: cfb0ef3108d49cc4562d5810b0a9af60

V: 4c89af496176b728ed1e2ea8ba27f5a4

The total number of one bits in the 256-bit seed is 124, or a fraction of 0.48,
which is reassuringly close to the ideal of 0.5.

For the OFB PRNG, Table 8.2 shows the first eight output blocks (1024 bits)
with two rough measures of security. The second column shows the fraction of one
bits in each 128-bit block. This corresponds to one of the NIST tests. The results
indicate that the output is split roughly equally between zero and one bits. The
third column shows the fraction of bits that match between adjacent blocks. If this

Figure 8.4 PRNG Mechanisms Based on Block Ciphers

(a) CTR mode

V

Encrypt

Pseudorandom bits

K

1

+

(b) OFB mode

V

Encrypt

Pseudorandom bits

K

Hiva-Network.Com

8.3 / PSEUDORANDOM NUMBER GENERATION USING A BLOCK CIPHER 263

Output Block
Fraction of
One Bits

Fraction of Bits
that Match with
Preceding Block

1786f4c7ff6e291dbdfdd90ec3453176 0.57 —
5e17b22b14677a4d66890f87565eae64 0.51 0.52
fd18284ac82251dfb3aa62c326cd46cc 0.47 0.54
c8e545198a758ef5dd86b41946389bd5 0.50 0.44
fe7bae0e23019542962e2c52d215a2e3 0.47 0.48
14fdf5ec99469598ae0379472803accd 0.49 0.52
6aeca972e5a3ef17bd1a1b775fc8b929 0.57 0.48
f7e97badf359d128f00d9b4ae323db64 0.55 0.45

Table 8.2 Example Results for PRNG Using OFB

Output Block
Fraction of
One Bits

Fraction of Bits
that Match with
Preceding Block

1786f4c7ff6e291dbdfdd90ec3453176 0.57 —

60809669a3e092a01b463472fdcae420 0.41 0.41

d4e6e170b46b0573eedf88ee39bff33d 0.59 0.45

5f8fcfc5deca18ea246785d7fadc76f8 0.59 0.52

90e63ed27bb07868c753545bdd57ee28 0.53 0.52

0125856fdf4a17f747c7833695c52235 0.50 0.47
f4be2d179b0f2548fd748c8fc7c81990 0.51 0.48
1151fc48f90eebac658a3911515c3c66 0.47 0.45

Table 8.3 Example Results for PRNG Using CTR

number differs substantially from 0.5, that suggests a correlation between blocks,
which could be a security weakness. The results suggest no correlation.

Table 8.3 shows the results using the same key and V values for CTR mode.
Again, the results are favorable.

ANSI X9.17 PRNG

One of the strongest (cryptographically speaking) PRNGs is specified in ANSI
X9.17. A number of applications employ this technique, including financial security
applications and PGP (the latter described in Chapter 19).

Figure 8.5 illustrates the algorithm, which makes use of triple DES for encryp-
tion. The ingredients are as follows.

■ Input: Two pseudorandom inputs drive the generator. One is a 64-bit represen-
tation of the current date and time, which is updated on each number genera-
tion. The other is a 64-bit seed value; this is initialized to some arbitrary value
and is updated during the generation process.

■ Keys: The generator makes use of three triple DES encryption modules. All
three make use of the same pair of 56-bit keys, which must be kept secret and
are used only for pseudorandom number generation.

264 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

■ Output: The output consists of a 64-bit pseudorandom number and a 64-bit
seed value.

Let us define the following quantities.

DTi Date/time value at the beginning of ith generation stage
Vi Seed value at the beginning of ith generation stage
Ri Pseudorandom number produced by the ith generation stage
K1, K2 DES keys used for each stage

Then

Ri = EDE([K1, K2], [Vi⊕ EDE([K1, K2], DTi)])
Vi+ 1 = EDE([K1, K2], [Ri⊕ EDE([K1, K2], DTi)])

where EDE([K1, K2], X) refers to the sequence encrypt-decrypt-encrypt using two-
key triple DES to encrypt X.

Several factors contribute to the cryptographic strength of this method. The
technique involves a 112-bit key and three EDE encryptions for a total of nine DES
encryptions. The scheme is driven by two independent inputs, the date and time
value, and a seed produced by the generator that is distinct from the pseudorandom
number produced by the generator. Thus, the amount of material that must be com-
promised by an opponent appears to be overwhelming. Even if a pseudorandom
number Ri were compromised, it would be impossible to deduce the Vi+ 1 from the
Ri, because an additional EDE operation is used to produce the Vi+ 1.

NIST CTR_DRBG

We now look more closely at the details of the PRNG defined in NIST SP 800-90A
based on the CTR mode of operation. The PRNG is referred to as CTRDRBG
(counter mode–deterministic random bit generator). CTR_DRBG is widely imple-
mented and is part of the hardware random number generator implemented on all
recent Intel processor chips (discussed in Section 8.6).

Figure 8.5 ANSI X9.17 Pseudorandom Number Generator

EDE

EDE

EDE

K1, K2

DTi

Vi

Vi+1

Ri

8.3 / PSEUDORANDOM NUMBER GENERATION USING A BLOCK CIPHER 265

The DRBG assumes that an entropy source is available to provide random
bits. Typically, the entropy source will be a TRNG based on some physical source.
Other sources are possible if they meet the required entropy measure of the appli-
cation. Entropy is an information theoretic concept that measures unpredictability,
or randomness; see Appendix F for details. The encryption algorithm used in the
DRBG may be 3DES with three keys or AES with a key size of 128, 192, or 256 bits.

Four parameters are associated with the algorithm:

■ Output block length (outlen): Length of the output block of the encryption
algorithm.

■ Key length (keylen): Length of the encryption key.

■ Seed length (seedlen): The seed is a string of bits that is used as input to a
DRBG mechanism. The seed will determine a portion of the internal state of
the DRBG, and its entropy must be sufficient to support the security strength
of the DRBG. seedlen = outlen + keylen.

■ Reseed interval (reseed_interval): Length of the encryption key. It is the maxi-
mum number of output blocks generated before updating the algorithm with
a new seed.

Table 8.4 lists the values specified in SP 800-90A for these parameters.

INITIALIZE Figure 8.6 shows the two principal functions that comprise CTR_DRBG.
We first consider how CTR_DRBG is initialized, using the initialize and update
function (Figure 8.6a). Recall that the CTR block cipher mode requires both an
encryption key K and an initial counter value, referred to in SP 800-90A as the
counter V. The combination of K and V is referred to as the seed. To start the
DRGB operation, initial values for K and V are needed, and can be chosen arbi-
trarily. As an example, the Intel Digital Random Number Generator, discussed in
Section 8.6, uses the values K = 0 and V = 0. These values are used as param-
eters for the CTR mode of operation to produce at least seedlen bits. In addition,
exactly seedlen bits must be supplied from what is referred to as an entropy source.
Typically, the entropy source would be some form of TRNG.

With these inputs, the CTR mode of encryption is iterated to produce a
sequence of output blocks, with V incremented by 1 after each encryption. The pro-
cess continues until at least seedlen bits have been generated. The leftmost seedlen
bits of output are then XORed with the seedlen entropy bits to produce a new seed.
In turn, the leftmost keylen bits of the seed form the new key and the rightmost
outlen bits of the seed form the new counter value V.

3DES AES-128 AES-192 AES-256

outlen 64 128 128 128
keylen 168 128 192 256
seedlen 232 256 320 384
reseed_interval …232 …248 …248 …248

Table 8.4 CTR_DRBG Parameters

266 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

GENERATE Once values of Key and V are obtained, the DRBG enters the generate
phase and is able to generate pseudorandom bits, one output block at a time
(Figure 8.6b). The encryption function is iterated to generate the number of pseu-
dorandom bits desired. Each iteration uses the same encryption key. The counter
value V is incremented by 1 for each iteration.

UPDATE To enhance security, the number of bits generated by any PRNG should be
limited. CTR_DRGB uses the parameter reseed_interval to set that limit. During the
generate phase, a reseed counter is initialized to 1 and then incremented with each

Figure 8.6 CTR_DRBG Functions

Encrypt

Iterate1

V

Entropy
source

1st
time +

Key

B0

Key

(a) Initialize and update function

(b) Generate function

Bi

V

Key V

Encrypt

Iterate1

+

8.4 / STREAM CIPHERS 267

iteration (each production of an output block). When the reseed counter reaches
reseed_interval, the update function is invoked (Figure 8.6a). The update function
is the same as the initialize function. In the update case, the Key and V values last
used by the generate function serve as the input parameters to the update function.
The update function takes seedlen new bits from an entropy source and produces a
new seed (Key, V). The generate function can then resume production of pseudo-
random bits. Note that the result of the update function is to change both the Key
and V values used by the generate function.

8.4 STREAM CIPHERS

A typical stream cipher encrypts plaintext one byte at a time, although a stream
cipher may be designed to operate on one bit at a time or on units larger than a byte
at a time. Figure 8.7 is a representative diagram of stream cipher structure. In this
structure, a key is input to a pseudorandom bit generator that produces a stream
of 8-bit numbers that are apparently random. The output of the generator, called
a keystream, is combined one byte at a time with the plaintext stream using the
bitwise exclusive-OR (XOR) operation. For example, if the next byte generated by
the generator is 01101100 and the next plaintext byte is 11001100, then the resulting
ciphertext byte is

11001100 plaintext
⊕ 01101100 key stream

10100000 ciphertext

Decryption requires the use of the same pseudorandom sequence:

10100000 ciphertext
⊕ 01101100 key stream

11001100 plaintext

Figure 8.7 Stream Cipher Diagram

Pseudorandom byte
generator

(key stream generator)

Plaintext
byte stream

M

Key
K

Key
K

k
Plaintext

byte stream
M

Ciphertext
byte stream

CENCRYPTION

Pseudorandom byte
generator

(key stream generator)

DECRYPTION

k

268 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

The stream cipher is similar to the one-time pad discussed in Chapter 3. The
difference is that a one-time pad uses a genuine random number stream, whereas a
stream cipher uses a pseudorandom number stream.

[KUMA97] lists the following important design considerations for a stream cipher.

1. The encryption sequence should have a large period. A pseudorandom num-
ber generator uses a function that produces a deterministic stream of bits that
eventually repeats. The longer the period of repeat the more difficult it will be
to do cryptanalysis. This is essentially the same consideration that was discussed
with reference to the Vigenère cipher, namely that the longer the keyword
the more difficult the cryptanalysis.

2. The keystream should approximate the properties of a true random number
stream as close as possible. For example, there should be an approximately
equal number of 1s and 0s. If the keystream is treated as a stream of bytes,
then all of the 256 possible byte values should appear approximately equally
often. The more random-appearing the keystream is, the more randomized the
ciphertext is, making cryptanalysis more difficult.

3. Note from Figure 8.7 that the output of the pseudorandom number genera-
tor is conditioned on the value of the input key. To guard against brute-force
attacks, the key needs to be sufficiently long. The same considerations that
apply to block ciphers are valid here. Thus, with current technology, a key
length of at least 128 bits is desirable.

With a properly designed pseudorandom number generator, a stream cipher
can be as secure as a block cipher of comparable key length. A potential advantage
of a stream cipher is that stream ciphers that do not use block ciphers as a building
block are typically faster and use far less code than do block ciphers. The example
in this chapter, RC4, can be implemented in just a few lines of code. In recent years,
this advantage has diminished with the introduction of AES, which is quite efficient
in software. Furthermore, hardware acceleration techniques are now available for
AES. For example, the Intel AES Instruction Set has machine instructions for one
round of encryption and decryption and key generation. Using the hardware in-
structions results in speedups of about an order of magnitude compared to pure
software implementations [XU10].

One advantage of a block cipher is that you can reuse keys. In contrast, if two
plaintexts are encrypted with the same key using a stream cipher, then cryptanalysis
is often quite simple [DAWS96]. If the two ciphertext streams are XORed together,
the result is the XOR of the original plaintexts. If the plaintexts are text strings,
credit card numbers, or other byte streams with known properties, then cryptanaly-
sis may be successful.

For applications that require encryption/decryption of a stream of data, such as
over a data communications channel or a browser/Web link, a stream cipher might
be the better alternative. For applications that deal with blocks of data, such as file
transfer, email, and database, block ciphers may be more appropriate. However,
either type of cipher can be used in virtually any application.

A stream cipher can be constructed with any cryptographically strong PRNG,
such as the ones discussed in Sections 8.2 and 8.3. In the next section, we look at a
stream cipher that uses a PRNG designed specifically for the stream cipher.

8.5 / RC4 269

8.5 RC4

RC4 is a stream cipher designed in 1987 by Ron Rivest for RSA Security. It is a
variable key size stream cipher with byte-oriented operations. The algorithm is
based on the use of a random permutation. Analysis shows that the period of the
cipher is overwhelmingly likely to be greater than 10100 [ROBS95a]. Eight to sixteen
machine operations are required per output byte, and the cipher can be expected
to run very quickly in software. RC4 is used in the WiFi Protected Access (WPA)
protocol that are part of the IEEE 802.11 wireless LAN standard. It is optional for
use in Secure Shell (SSH) and Kerberos. RC4 was kept as a trade secret by RSA
Security. In September 1994, the RC4 algorithm was anonymously posted on the
Internet on the Cypherpunks anonymous remailers list.

The RC4 algorithm is remarkably simple and quite easy to explain.
A variable-length key of from 1 to 256 bytes (8 to 2048 bits) is used to initialize a
256-byte state vector S, with elements S[0],S[1], . . . ,S[255]. At all times, S contains
a permutation of all 8-bit numbers from 0 through 255. For encryption and decryp-
tion, a byte k (see Figure 8.7) is generated from S by selecting one of the 255 entries
in a systematic fashion. As each value of k is generated, the entries in S are once
again permuted.

Initialization of S

To begin, the entries of S are set equal to the values from 0 through 255 in ascending
order; that is, S[0] = 0, S[1] = 1, c , S[255] = 255. A temporary vector, T, is also
created. If the length of the key K is 256 bytes, then K is transferred to T. Otherwise,
for a key of length keylen bytes, the first keylen elements of T are copied from K,
and then K is repeated as many times as necessary to fill out T. These preliminary
operations can be summarized as

/* Initialization */
for i = 0 to 255 do
S[i] = i;
T[i] = K[i mod keylen];

Next we use T to produce the initial permutation of S. This involves starting
with S[0] and going through to S[255], and for each S[i], swapping S[i] with another
byte in S according to a scheme dictated by T[i]:

/* Initial Permutation of S */
j = 0;
for i = 0 to 255 do

j = (j + S[i] + T[i]) mod 256;
Swap (S[i], S[j]);

Because the only operation on S is a swap, the only effect is a permutation.
S still contains all the numbers from 0 through 255.

270 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

Stream Generation

Once the S vector is initialized, the input key is no longer used. Stream generation
involves cycling through all the elements of S[i], and for each S[i], swapping S[i]
with another byte in S according to a scheme dictated by the current configuration
of S. After S[255] is reached, the process continues, starting over again at S[0]:

/* Stream Generation */
i, j = 0;
while (true)
i = (i + 1) mod 256;
j = (j + S[i]) mod 256;
Swap (S[i], S[j]);
t = (S[i] + S[j]) mod 256;
k = S[t];

To encrypt, XOR the value k with the next byte of plaintext. To decrypt, XOR
the value k with the next byte of ciphertext.

Figure 8.8 illustrates the RC4 logic.

Figure 8.8 RC4

25525425343210S

T

S

(a) Initial state of S and T

(b) Initial permutation of S

Swap

T

K

T[i]

j = j + S[i] + T[i]

t = S[i] + S[j]

]j[S]i[S

Keylen

i

S

(c) Stream generation

Swap

j = j + S[i]

]t[S]j[S]i[S

k

i

8.6 / TRUE RANDOM NUMBER GENERATORS 271

Strength of RC4

A number of papers have been published analyzing methods of attacking RC4 (e.g.,
[KNUD98], [FLUH00], [MANT01]). None of these approaches is practical against
RC4 with a reasonable key length, such as 128 bits. A more serious problem is reported
in [FLUH01]. The authors demonstrate that the WEP protocol, intended to provide
confidentiality on 802.11 wireless LAN networks, is vulnerable to a particular attack
approach. In essence, the problem is not with RC4 itself but the way in which keys are
generated for use as input to RC4. This particular problem does not appear to be rele-
vant to other applications using RC4 and can be remedied in WEP by changing the way
in which keys are generated. This problem points out the difficulty in designing a secure
system that involves both cryptographic functions and protocols that make use of them.

More recently, [PAUL07] revealed a more fundamental vulnerability in the
RC4 key scheduling algorithm that reduces the amount of effort to discover the
key. Recent cryptanalysis results [ALFA13] exploit biases in the RC4 keystream to
recover repeatedly encrypted plaintexts. As a result of the discovered weaknesses,
particularly those reported in [ALFA13], the IETF issued RFC 7465 prohibiting the
use of RC4 in TLS (Prohibiting RC4 Cipher Suites, February 2015). In its latest TLS
guidelines, NIST also prohibited the use of RC4 for government use (SP 800-52,
Guidelines for the Selection, Configuration, and Use of Transport Layer Security
(TLS) Implementations, September 2013).

8.6 TRUE RANDOM NUMBER GENERATORS

Entropy Sources

A true random number generator (TRNG) uses a nondeterministic source to pro-
duce randomness. Most operate by measuring unpredictable natural processes, such
as pulse detectors of ionizing radiation events, gas discharge tubes, and leaky capac-
itors. Intel has developed a commercially available chip that samples thermal noise
by sampling the output of a coupled pair of inverters. LavaRnd is an open source
project for creating truly random numbers using inexpensive cameras, open source
code, and inexpensive hardware. The system uses a saturated CCD in a light-tight
can as a chaotic source to produce the seed. Software processes the result into truly
random numbers in a variety of formats.

RFC 4086 lists the following possible sources of randomness that, with care,
easily can be used on a computer to generate true random sequences.

■ Sound/video input: Many computers are built with inputs that digitize some
real-world analog source, such as sound from a microphone or video input
from a camera. The “input” from a sound digitizer with no source plugged in or
from a camera with the lens cap on is essentially thermal noise. If the system
has enough gain to detect anything, such input can provide reasonably high
quality random bits.

■ Disk drives: Disk drives have small random fluctuations in their rotational
speed due to chaotic air turbulence [JAKO98]. The addition of low-level disk
seek-time instrumentation produces a series of measurements that contain this

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272 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

randomness. Such data is usually highly correlated, so significant processing is
needed. Nevertheless, experimentation a decade ago showed that, with such
processing, even slow disk drives on the slower computers of that day could
easily produce 100 bits a minute or more of excellent random data.

There is also an online service (random.org), which can deliver random
sequences securely over the Internet.

Comparison of PRNGs and TRNGs

Table 8.5 summarizes the principal differences between PRNGs and TRNGs.
PRNGs are efficient, meaning they can produce many numbers in a short time, and
deterministic, meaning that a given sequence of numbers can be reproduced at a
later date if the starting point in the sequence is known. Efficiency is a nice char-
acteristic if your application needs many numbers, and determinism is handy if you
need to replay the same sequence of numbers again at a later stage. PRNGs are
typically also periodic, which means that the sequence will eventually repeat itself.
While periodicity is hardly ever a desirable characteristic, modern PRNGs have a
period that is so long that it can be ignored for most practical purposes.

TRNGs are generally rather inefficient compared to PRNGs, taking consid-
erably longer time to produce numbers. This presents a difficulty in many applica-
tions. For example, cryptography system in banking or national security might need
to generate millions of random bits per second. TRNGs are also nondeterministic,
meaning that a given sequence of numbers cannot be reproduced, although the same
sequence may of course occur several times by chance. TRNGs have no period.

Conditioning2

A TRNG may produce an output that is biased in some way, such as having more
ones than zeros or vice versa. More generally, NIST SP 800-90B defines a random
process as biased with respect to an assumed discrete set of potential outcomes
(i.e., possible output values) if some of those outcomes have a greater probability
of occurring than do others. For example, a physical source such as electronic noise
may contain a superposition of regular structures, such as waves or other periodic
phenomena, which may appear to be random, yet are determined to be non-random
using statistical tests.

2 The reader unfamiliar with the concepts of entropy and min-entropy should read Appendix F before
proceeding.

Pseudorandom Number
Generators

True Random Number
Generators

Efficiency Very efficient Generally inefficient
Determinism Deterministic Nondeterministic
Periodicity Periodic Aperiodic

Table 8.5 Comparison of PRNGs and TRNGs

8.6 / TRUE RANDOM NUMBER GENERATORS 273

In addition to bias, another concept used by SP 800-98B is that of entropy rate.
SP 800-90B defines entropy rate as the rate at which a digitized noise source (or
entropy source) provides entropy; it is computed as the assessed amount of entropy
provided by a bit string output from the source, divided by the total number of
bits in the bit string (yielding assessed bits of entropy per output bit). This will be
a value between 0 (no entropy) and 1 (full entropy). Entropy rate is a measure
of the randomness or unpredictability of a bit string. Another way of express-
ing it is that the entropy rate is k/n for a random source of length n bits and min-
entropy k. Min-entropy is a measure of the number of random bits and is explained
in Appendix F. In essence, a block of bits or a bit stream that is unbiased, and in
which each bit and each group of bits is independent of all other bits and groups of
bits will have an entropy rate of 1.

For hardware sources of random bits, the recommended approach is to assume
that there may be bias and/or an entropy rate of less than 1 and to apply techniques
to further “randomize” the bits. Various methods of modifying a bit stream for this
purpose have been developed. These are referred to as conditioning algorithms or
deskewing algorithms.

Typically, conditioning is done by using a cryptographic algorithm to “ scramble”
the random bits so as to eliminate bias and increase entropy. The two most common
approaches are the use of a hash function or a symmetric block cipher.

HASH FUNCTION As we describe in Chapter 11, a hash function produces an n-bit
output from an input of arbitrary length. A simple way to use a hash function for
conditioning is as follows. Blocks of m input bits, with m Ú n, are passed through
the hash function and the n output bits are used as random bits. To generate a
stream of random bits, successive input blocks pass through the hash function to
produce successive hashed output blocks.

Operating systems typically provide a built-in mechanism for generating ran-
dom numbers. For example, Linux uses four entropy sources: mouse and keyboard
activity, disk I/O operations, and specific interrupts. Bits are generated from these
four sources and combined in a pooled buffer. When random bits are needed, the
appropriate number of bits are read from the buffer and passed through the SHA-1
hash function [GUTT06].

A more complex approach is the hash derivation function specified in
SP800-90A. Hash_df can be defined as follows:

Parameters:

input_string: The string to be hashed.

outlen: Output length.

no_of_bits_to_return: The number of bits to be returned by Hash_df. The maxi-
mum length (max_number_of_bits) is implementation dependent, but shall be
less than or equal to (255 * outlen). no_of_bits_to_return is represented as a
32-bit integer.

requested_bits: The result of performing the Hash_df.

274 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

Hash_df Process:

1. temp = the Null string

2. len = l no_of_bits_to_return
outlen

m
3. counter = 0x01 Comment: An 8-bit binary value representing the integer “1”.
4. For i = 1 to len do Comment: In 4.1, no_of_bits_to_return is used as a 32-bit

string.

4.1. temp = temp } Hash (counter } no_of_bits_to_return } input_string).
4.2. counter = counter + 1.

5. requested_bits = leftmost (temp, no_of_bits_to_return).
6. Return (SUCCESS, requested_bits).

This algorithm takes an input block of bits of arbitrary length and returns the
requested number of bits, which may be up to 255 times as long as the hash output
length.

The reader may be uneasy that the output consists of hashed blocks in which
the input to the hash function for each block is the same input string and differs
only by the value of the counter. However, cryptographically strong hash functions,
such as the SHA family, provide excellent diffusion (as defined in Chapter 4) so that
change in the counter value results in dramatically different outputs.

BLOCK CIPHER Instead of a hash function, a block cipher such as AES can be
used to scramble the TRNG bits. Using AES, a simple approach would be to take
128-bit blocks of TRNG bits and encrypt each block with AES and some arbitrary
key. SP 800-90B outlines an approach similar to the hash_df function described pre-
viously. The Intel implementation discussed subsequently provides an example of
using AES for conditioning.

Health Testing

Figure 8.9 provides a general model for a nondeterministic random bit generator.
A hardware noise source produces a true random output. This is digitized to pro-
duce true, or nondeterministic, source of bits. This bit source then passes through a
conditioning module to mitigate bias and maximize entropy.

Figure 8.9 also shows a health-testing module, which is used on the outputs
of both the digitizer and conditioner. In essence, health testing is used to validate
that the noise source is working as expected and that the conditioning module is
produced output with the desired characteristics. Both forms of health testing are
recommended by SP 800-90B.

HEALTH TESTS ON THE NOISE SOURCE The nature of the health testing of the noise
source depends strongly on the technology used to produce noise. In general, we
can assume that the digitized output of the noise source will exhibit some bias. Thus,
the traditional statistical tests, such as those defined in SP 800-22 and discussed in
Section 8.1, are not useful for monitoring the noise source, because the noise source

8.6 / TRUE RANDOM NUMBER GENERATORS 275

Figure 8.9 NRBG Model

Nondetermistic bit source

Noise source

Digitization

Conditioning

A

B

Output

Health
testing

NONDETERMINISTIC
RANDOM

BIT GENERATOR

is likely to always fail. Rather, the tests on the noise source need to be tailored to
the expected statistical behavior of the correctly operating noise source. The goal
is not to determine if the source is unbiased, which it isn’t, but if it is operating
as expected.

SP 800-90B specifies that continuous tests be done on digitized samples
obtained from the noise source (point A in Figure 8.9). The purpose is to test for
variability. More specifically, the purpose is to determine if the noise source is pro-
ducing at the expected entropy rate. SP 800-909B mandates the use of two tests: the
Repetition Count Test and the Adaptive Proportion Test.

The Repetition Count Test is designed to quickly detect a catastrophic failure
that causes the noise source to become “stuck” on a single output value for a long
time. For this test, it is assumed that a given noise source is assessed to have a given
min-entropy value of H. The entropy is expressed as the amount of entropy per sam-
ple, where a sample could be a single bit or some block of bits of length n. With an
assessed value of H, it is straightforward to calculate the probability that a sequence
of C consecutive samples will yield identical sample values. For example, a noise
source with one bit of min-entropy per sample has no more than a 1/2 probability
of repeating some sample value twice in a row, no more than 1/4 probability of
repeating some sample value three times in a row, and in general, no more than
(1/2)C - 1 probability of repeating some sample value C times in a row. To generalize,
for a noise source with H bits of min-entropy per sample, we have:

Pr[C identical samples in a row] … (2-H)(C - 1)

The Repetition Count Test involves looking for consecutive identical sam-
ples. If the count reaches some cutoff value C, then an error condition is raised.
To determine the value of C used in the test, the test must be configured with

276 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

a parameter W, which is the acceptable false-positive probability associated with
an alarm triggered by C repeated sample values. To avoid false positives, W should
be set at some very small number greater than 0. Given W, we can now determine
the value of C. Specifically, we want C to be the smallest number that satisfies the
equation W … (2-H)(C - 1). Reworking terms, this gives us a value of:

C = l 1 + - log(W)
H

m
For example, for W = 2-30, an entropy source with H = 7.3 bits per sample

would have a cutoff value C of l 1 + 30
7.3
m = 6.

The Repetition Count Test starts by recording a sample value and then count-
ing the number of repetitions of the same value. If the counter reaches the cutoff
value C, an error is reported. If a sample value is encountered that differs from the
preceding sample, then the counter is reset to 1 and the algorithm starts over.

The Adaptive Proportion Test is designed to detect a large loss of entropy,
such as might occur as a result of some physical failure or environmental change
affecting the noise source. The test continuously measures the local frequency of
occurrence of some sample value in a sequence of noise source samples to determine
if the sample occurs too frequently.

The test starts by recording a sample value and then observes N successive
sample values. If the initial sample value is observed at least C times, then an error
condition is reported. SP 800-90B recommends that a probability of a false positive
of W = 2-30 be used for the test and provides guidance on the selection of values
for N and C.

HEALTH TESTS ON THE CONDITIONING FUNCTION SP 800-90B specifies that health
tests should also be applied to the output of the conditioning component (point B
in Figure 8.9), but does not indicate which tests to use. The purpose of the health
tests on the conditioning component is to assure that the output behaves as a true
random bit stream. Thus, it is reasonable to use the tests for randomness defined in
SP 800-22, and described in Section 8.1.

Intel Digital Random Number Generator

As was mentioned, TRNGs have traditionally been used only for key generation
and other applications where only a small number of random bits were required.
This is because TRNGs have generally been inefficient, with a low bit rate of
random bit production.

The first commercially available TRNG that achieves bit production rates
comparable with that of PRNGs is the Intel digital random number generator
(DRNG) [TAYL11, MECH14], offered on new multicore chips since May 2012.3

3It is unfortunate that Intel chose the acronym DRNG for an NRBG. It confuses with DRBG, which is
a pseudorandom number bit generator.

8.6 / TRUE RANDOM NUMBER GENERATORS 277

Two notable aspects of the DRNG:

1. It is implemented entirely in hardware. This provides greater security than a
facility that includes a software component. A hardware-only implementa-
tion should also be able to achieve greater computation speed than a software
module.

2. The entire DRNG is on the same multicore chip as the processors. This elimi-
nates the I/O delays found in other hardware random number generators.

DRNG HARDWARE ARCHITECTURE Figure 8.10 shows the overall structure of the
DRNG. The first stage of the DRNG generates random numbers from thermal
noise. The heart of the stage consists of two inverters (NOT gates), with the output
of each inverter connected to the input of the other. Such an arrangement has two
stable states, with one inverter having an output of logical 1 and the other having an
output of logical 0. The circuit is then configured so that both inverters are forced
to have the same indeterminate state (both inputs and both outputs at logical 1) by
clock pulses. Random thermal noise within the inverters soon jostles the two invert-
ers into a mutually stable state. Additional circuitry is intended to compensate for
any biases or correlations. This stage is capable, with current hardware, of generat-
ing random bits at a rate of 4 Gbps.

Figure 8.10 Intel Processor Chip with Random Number Generator

Hardware
AES-CBC-
MAC based
conditioner

Digital Random Number Generator

Processor
chip

Hardware
SP 800-90A
AES-CTR

based
DRBGHardware

entropy
source

RDSEED
instructionCore 0

Core N–1 RDSEEDinstruction

RDRAND
instruction

RDRAND
instruction

Hardware
SP 800-
90B & C
ENRNG

278 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

The output of the first stage is generated 512 bits at a time. To assure that
the bit stream does not have skew or bias, a conditioner randomizes its input using
a cryptographic function. In this case, the function is referred to as CBC-MAC or
CMAC, as specified in NIST SP 800-38B. In essence, CMAC encrypts its input using
the cipher block chaining (CBC) mode (Figure 8.4) and outputs the final block.
We examine CMAC in detail in Chapter 12. The output of this stage is generated
256 bits at a time and is intended to exhibit true randomness with no skew or bias.

While the hardware’s circuitry generates random numbers from thermal noise
much more quickly than its predecessors, it is still not fast enough for some of to-
day’s computing requirements. To enable the DRNG to generate random numbers
as quickly as a software DRBG, and also maintain the high quality of the random
numbers, a third stage is added. This stage uses the 256-bit random numbers to
seed a cryptographically secure DRBG that creates 128-bit numbers. From one
256-bit seed, the DRBG can output many pseudorandom numbers, exceeding the
3-Gbps rate of the entropy source. An upper bound of 511 128-bit samples can
be generated per seed. The algorithm used for this stage is CTR_DRBG, described
in Section 8.3.

The output of the PRNG stage is available to each of the cores on the chip via
the RDRAND instruction. RDRAND retrieves a 16-, 32-, or 64-bit random value
and makes it available in a software-accessible register.

Preliminary data from a pre-production sample on a system with a third
generation Intel® Core™ family processor produced the following performance
[INTE12]: up to 70 million RDRAND invocations per second, and a random data
production rate of over 4 Gbps.

The output of the conditioner is also made available to another module,
known as an enhanced nondeterministic random number generator (ENRNG) that
provides random numbers that can be used as seeds for various cryptographic algo-
rithms. The ENRNG is compliant with specifications in SP 800-90B and 900-90C.
The output of the ENRNG stage is available to each of the cores on the chip via
the RDSEED instruction. RDSEED retrieves a hardware-generated random seed
value from the ENRNG and stores it in the destination register given as an argu-
ment to the instruction.

DRNG LOGICAL STRUCTURE Figure 8.11 provides a simplified view of the logical
flow of the Intel DRBG. As was described, the heart of the hardware entropy source
is a pair of inverters that feed each other. Two transistors, driven by the same clock,
force the inputs and outputs of both inverters to the logical 1 state. Because this is
an unstable state, thermal noise will cause the configuration to settle randomly into
a stable state with either Node A at logical 1 and Node B at logical 0, or the reverse.
Thus the module generates random bits at the clock rate.

The output of the entropy source is collected 512 bits at a time and used to
feed to two CBC hardware implementations using AES encryption. Each imple-
mentation takes two blocks of 128 bits of “plaintext” and encrypts using the CBC
mode. The output of the second encryption is retained. For both CBC modules, an
all-zeros key is used initially. Subsequently, the output of the PRNG stage is fed
back to become the key for the conditioner stage.

8.6 / TRUE RANDOM NUMBER GENERATORS 279

Figure 8.11 Intel DRNG Logical Structure

EncryptEncrypt

128 bits 128 bits

Clock

Transistor 1 Transistor 2

Inverters

Node A Node B

128 bits 128 bits

128 bits

Key V

K K K K

Encrypt Encrypt

Pseudorandom
bits

128 bits

101st
time

+

256 bits

EncryptEncrypt

128 bits

1

+

Hardware
entropy
source

AES CBC
Mac-based
conditioner

AES-CTR-
based
PRNG

K = 0

The output of the conditioner stage consists of 256 bits. This block is provided
as input to the update function of the DRGB stage. The update function is initial-
ized with the all-zeros key and the counter value 0. The function is iterated twice
to produce a 256-block, which is then XORed with the input from the conditioner
stage. The results are used as the 128-bit key and the 128-bit seed for the generate
function. The generate function produces pseudorandom bits in 128-bit blocks.

280 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

8.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

backward unpredictability
Blum Blum Shub generator
deskewing
entropy source
forward unpredictability
keystream
linear congruential generator

pseudorandom function
(PRF)

pseudorandom number
generator (PRNG)

randomness
RC4
seed

stream cipher
skew
true random number

generator (TRNG)
unpredictability

Review Questions

8.1 List two criteria to validate the randomness of a sequence of numbers.
8.2 What is ANSI X9.17 PRNG?
8.3 What is the difference between a one-time pad and a stream cipher?
8.4 List a few applications of stream ciphers and block ciphers.

Problems

8.1 If we take the linear congruential algorithm with an additive component of 0,

Xn + 1 = (aXn) mod m

Then it can be shown that if m is prime and if a given value of a produces the maxi-
mum period of m - 1, then ak will also produce the maximum period, provided that
k is less than m and that k and m - 1 are relatively prime. Demonstrate this by using
X0 = 1 and m = 31 and producing the sequences for ak = 3, 32, 33, and 34.

8.2 a. What is the maximum period obtainable from the following generator?

Xn + 1 = (aXn) mod 24

b. What should be the value of a?
c. What restrictions are required on the seed?

8.3 You may wonder why the modulus m = 231 - 1 was chosen for the linear congruen-
tial method instead of simply 231, because this latter number can be represented with
no additional bits and the mod operation should be easier to perform. In general, the
modulus 2k - 1 is preferable to 2k. Why is this so?

8.4 With the linear congruential algorithm, a choice of parameters that provides a full
period does not necessarily provide a good randomization. For example, consider the
following two generators:

Xn + 1 = (11Xn) mod 13

Xn + 1 = (2Xn) mod 13

Write out the two sequences to show that both are full period. Which one appears
more random to you?

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8.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 281

8.5 In any use of pseudorandom numbers, whether for encryption, simulation, or statisti-
cal design, it is dangerous to trust blindly the random number generator that happens
to be available in your computer’s system library. [PARK88] found that many con-
temporary textbooks and programming packages make use of flawed algorithms for
pseudorandom number generation. This exercise will enable you to test your system.

The test is based on a theorem attributed to Ernesto Cesaro (see [KNUT98] for a
proof), which states the following: Given two randomly chosen integers, x and y, the
probability that gcd(x, y) = 1 is 6/p2. Use this theorem in a program to determine
statistically the value of p. The main program should call three subprograms: the ran-
dom number generator from the system library to generate the random integers; a
subprogram to calculate the greatest common divisor of two integers using Euclid’s
Algorithm; and a subprogram that calculates square roots. If these latter two pro-
grams are not available, you will have to write them as well. The main program should
loop through a large number of random numbers to give an estimate of the afore-
mentioned probability. From this, it is a simple matter to solve for your estimate of p.

If the result is close to 3.14, congratulations! If not, then the result is probably low,
usually a value of around 2.7. Why would such an inferior result be obtained?

8.6 What RC4 key value will leave S unchanged during initialization? That is, after the
initial permutation of S, the entries of S will be equal to the values from 0 through 255
in ascending order.

8.7 RC4 has a secret internal state which is a permutation of all the possible values of the
vector S and the two indices i and j.
a. Using a straightforward scheme to store the internal state, how many bits are used?
b. Suppose we think of it from the point of view of how much information is repre-

sented by the state. In that case, we need to determine how may different states
there are, then take the log to base 2 to find out how many bits of information this
represents. Using this approach, how many bits would be needed to represent the
state?

8.8 Alice and Bob agree to communicate privately via email using a scheme based on
RC4, but they want to avoid using a new secret key for each transmission. Alice and
Bob privately agree on a 128-bit key k. To encrypt a message m, consisting of a string
of bits, the following procedure is used.
1. Choose a random 64-bit value v
2. Generate the ciphertext c = RC4(v }k)⊕ m
3. Send the bit string (v } c)

a. Suppose Alice uses this procedure to send a message m to Bob. Describe how
Bob can recover the message m from (v } c) using k.

b. If an adversary observes several values (v1 } c1), (v2 } c2), c transmitted
between Alice and Bob, how can he/she determine when the same key stream
has been used to encrypt two messages?

c. Approximately how many messages can Alice expect to send before the same
key stream will be used twice? Use the result from the birthday paradox
described in Appendix U.

d. What does this imply about the lifetime of the key k (i.e., the number of mes-
sages that can be encrypted using k)?

8.9 Suppose you have a true random bit generator where each bit in the generated stream
has the same probability of being a 0 or 1 as any other bit in the stream and that the
bits are not correlated; that is the bits are generated from identical independent dis-
tribution. However, the bit stream is biased. The probability of a 1 is 0.5 + 0 and the
probability of a 0 is 0.5 - 0, where 0 6 0 6 0.5. A simple conditioning algorithm is
as follows: Examine the bit stream as a sequence of nonoverlapping pairs. Discard all
00 and 11 pairs. Replace each 01 pair with 0 and each 10 pair with 1.

282 CHAPTER 8 / RANDOM BIT GENERATION AND STREAM CIPHERS

a. What is the probability of occurrence of each pair in the original sequence?
b. What is the probability of occurrence of 0 and 1 in the modified sequence?
c. What is the expected number of input bits to produce x output bits?
d. Suppose that the algorithm uses overlapping successive bit pairs instead of non-

overlapping successive bit pairs. That is, the first output bit is based on input bits 1
and 2, the second output bit is based on input bits 2 and 3, and so on. What can you
say about the output bit stream?

8.10 Another approach to conditioning is to consider the bit stream as a sequence of non-
overlapping groups of n bits each and output the parity of each group. That is, if a
group contains an odd number of ones, the output is 1; otherwise the output is 0.
a. Express this operation in terms of a basic Boolean function.
b. Assume, as in the preceding problem, that the probability of a 1 is 0.5 + 0. If each

group consists of 2 bits, what is the probability of an output of 1?
c. If each group consists of 4 bits, what is the probability of an output of 1?
d. Generalize the result to find the probability of an output of 1 for input groups of

n bits.
8.11 It is important to note that the Repetition Count Test described in Section 8.6 is not a

very powerful health test. It is able to detect only catastrophic failures of an entropy
source. For example, a noise source evaluated at 8 bits of min-entropy per sample
has a cutoff value of 5 repetitions to ensure a false-positive rate of approximately
once per four billion samples generated. If that noise source somehow failed to the
point that it was providing only 6 bits of min-entropy per sample, how many samples
would be expected to be needed before the Repetition Count Test would notice the
problem?

283

Public-Key Cryptography
and RSA

9.1 Principles of Public-Key Cryptosystems

Public-Key Cryptosystems
Applications for Public-Key Cryptosystems
Requirements for Public-Key Cryptography
Public-Key Cryptanalysis

9.2 The RSA Algorithm

Description of the Algorithm
Computational Aspects
The Security of RSA

9.3 Key Terms, Review Questions, and Problems

CHAPTER

PART THREE: ASYMMETRIC CIPHERS

284 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

The development of public-key, or asymmetric, cryptography is the greatest and per-
haps the only true revolution in the entire history of cryptography. From its earliest
beginnings to modern times, virtually all cryptographic systems have been based on
the elementary tools of substitution and permutation. After millennia of working with
algorithms that could be calculated by hand, a major advance in symmetric cryptogra-
phy occurred with the development of the rotor encryption/decryption machine. The
electromechanical rotor enabled the development of fiendishly complex cipher sys-
tems. With the availability of computers, even more complex systems were devised,
the most prominent of which was the Lucifer effort at IBM that culminated in the Data
Encryption Standard (DES). But both rotor machines and DES, although represent-
ing significant advances, still relied on the bread-and-butter tools of substitution and
permutation.

Public-key cryptography provides a radical departure from all that has gone be-
fore. For one thing, public-key algorithms are based on mathematical functions rather
than on substitution and permutation. More important, public-key cryptography is
asymmetric, involving the use of two separate keys, in contrast to symmetric encryp-
tion, which uses only one key. The use of two keys has profound consequences in the
areas of confidentiality, key distribution, and authentication, as we shall see.

Before proceeding, we should mention several common misconceptions con-
cerning public-key encryption. One such misconception is that public-key encryption
is more secure from cryptanalysis than is symmetric encryption. In fact, the security of
any encryption scheme depends on the length of the key and the computational work
involved in breaking a cipher. There is nothing in principle about either symmetric or
public-key encryption that makes one superior to another from the point of view of
resisting cryptanalysis.

A second misconception is that public-key encryption is a general-purpose tech-
nique that has made symmetric encryption obsolete. On the contrary, because of the
computational overhead of current public-key encryption schemes, there seems no
foreseeable likelihood that symmetric encryption will be abandoned. As one of the
inventors of public-key encryption has put it [DIFF88], “the restriction of public-key
cryptography to key management and signature applications is almost universally
accepted.”

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Present an overview of the basic principles of public-key cryptosystems.

◆ Explain the two distinct uses of public-key cryptosystems.

◆ List and explain the requirements for a public-key cryptosystem.

◆ Present an overview of the RSA algorithm.

◆ Understand the timing attack.

◆ Summarize the relevant issues related to the complexity of algorithms.

9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 285

Finally, there is a feeling that key distribution is trivial when using public-key
encryption, compared to the rather cumbersome handshaking involved with key dis-
tribution centers for symmetric encryption. In fact, some form of protocol is needed,
generally involving a central agent, and the procedures involved are not simpler nor
any more efficient than those required for symmetric encryption (e.g., see analysis in
[NEED78]).

This chapter and the next provide an overview of public-key cryptography. First,
we look at its conceptual framework. Interestingly, the concept for this technique was
developed and published before it was shown to be practical to adopt it. Next, we ex-
amine the RSA algorithm, which is the most important encryption/decryption algo-
rithm that has been shown to be feasible for public-key encryption. Other important
public-key cryptographic algorithms are covered in Chapter 10.

Much of the theory of public-key cryptosystems is based on number theory. If
one is prepared to accept the results given in this chapter, an understanding of number
theory is not strictly necessary. However, to gain a full appreciation of public-key
algorithms, some understanding of number theory is required. Chapter 2 provides the
necessary background in number theory.

Table 9.1 defines some key terms.

9.1 PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS

The concept of public-key cryptography evolved from an attempt to attack two of
the most difficult problems associated with symmetric encryption. The first problem
is that of key distribution, which is examined in some detail in Chapter 14.

As Chapter 14 discusses, key distribution under symmetric encryption requires
either (1) that two communicants already share a key, which somehow has been dis-
tributed to them; or (2) the use of a key distribution center. Whitfield Diffie, one

Asymmetric Keys
Two related keys, a public key and a private key, that are used to perform complementary operations, such as
encryption and decryption or signature generation and signature verification.

Public Key Certificate
A digital document issued and digitally signed by the private key of a Certification Authority that binds the
name of a subscriber to a public key. The certificate indicates that the subscriber identified in the certificate
has sole control and access to the corresponding private key.

Public Key (Asymmetric) Cryptographic Algorithm
A cryptographic algorithm that uses two related keys, a public key and a private key. The two keys have the
property that deriving the private key from the public key is computationally infeasible.

Public Key Infrastructure (PKI)
A set of policies, processes, server platforms, software and workstations used for the purpose of administer-
ing certificates and public-private key pairs, including the ability to issue, maintain, and revoke public key
certificates.

Source: Glossary of Key Information Security Terms, NIST IR 7298 [KISS06].

Table 9.1 Terminology Related to Asymmetric Encryption

286 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

of the discoverers of public-key encryption (along with Martin Hellman, both at
Stanford University at the time), reasoned that this second requirement negated the
very essence of cryptography: the ability to maintain total secrecy over your own
communication. As Diffie put it [DIFF88], “what good would it do after all to de-
velop impenetrable cryptosystems, if their users were forced to share their keys with
a KDC that could be compromised by either burglary or subpoena?”

The second problem that Diffie pondered, and one that was apparently un-
related to the first, was that of digital signatures. If the use of cryptography was to
become widespread, not just in military situations but for commercial and private
purposes, then electronic messages and documents would need the equivalent of
signatures used in paper documents. That is, could a method be devised that would
stipulate, to the satisfaction of all parties, that a digital message had been sent by a
particular person? This is a somewhat broader requirement than that of authentica-
tion, and its characteristics and ramifications are explored in Chapter 13.

Diffie and Hellman achieved an astounding breakthrough in 1976 [DIFF76 a, b]
by coming up with a method that addressed both problems and was radically different
from all previous approaches to cryptography, going back over four millennia.1

In the next subsection, we look at the overall framework for public-key cryp-
tography. Then we examine the requirements for the encryption/decryption algo-
rithm that is at the heart of the scheme.

Public-Key Cryptosystems

Asymmetric algorithms rely on one key for encryption and a different but related
key for decryption. These algorithms have the following important characteristic.

■ It is computationally infeasible to determine the decryption key given only
knowledge of the cryptographic algorithm and the encryption key.

In addition, some algorithms, such as RSA, also exhibit the following characteristic.

■ Either of the two related keys can be used for encryption, with the other used
for decryption.

A public-key encryption scheme has six ingredients (Figure 9.1a; compare
with Figure 3.1).

■ Plaintext: This is the readable message or data that is fed into the algorithm
as input.

■ Encryption algorithm: The encryption algorithm performs various transfor-
mations on the plaintext.

1Diffie and Hellman first publicly introduced the concepts of public-key cryptography in 1976. Hellman
credits Merkle with independently discovering the concept at that same time, although Merkle did not
publish until 1978 [MERK78]. In fact, the first unclassified document describing public-key distribution
and public-key cryptography was a 1974 project proposal by Merkle (http://merkle.com/1974). However,
this is not the true beginning. Admiral Bobby Inman, while director of the National Security Agency
(NSA), claimed that public-key cryptography had been discovered at NSA in the mid-1960s [SIMM93].
The first documented introduction of these concepts came in 1970, from the Communications-Electronics
Security Group, Britain’s counterpart to NSA, in a classified report by James Ellis [ELLI70]. Ellis re-
ferred to the technique as nonsecret encryption and describes the discovery in [ELLI99].

9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 287

■ Public and private keys: This is a pair of keys that have been selected so that if
one is used for encryption, the other is used for decryption. The exact transfor-
mations performed by the algorithm depend on the public or private key that
is provided as input.

■ Ciphertext: This is the encrypted message produced as output. It depends on
the plaintext and the key. For a given message, two different keys will produce
two different ciphertexts.

Figure 9.1 Public-Key Cryptography

Plaintext
input

Bobs's
public-key

ring

Transmitted
ciphertext

Plaintext
outputEncryption algorithm

(e.g., RSA)
Decryption algorithm

Joy

Mike

Mike Bob

Ted

Alice

Alice's public
key

Alice's private
key

(a) Encryption with public key

Plaintext
input

Transmitted
ciphertext

Plaintext
outputEncryption algorithm

(e.g., RSA)
Decryption algorithm

Bob's private
key

Bob

Bob's public
key

Alice's
public key

ring

Joy
Ted

(b) Encryption with private key

X

X

PUa

PUb

PRa

PRb

Y = E[PUa, X]

Y = E[PRb, X]

X =
D[PRa, Y]

X =
D[PUb, Y]

Alice

Bob Alice

288 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

■ Decryption algorithm: This algorithm accepts the ciphertext and the matching
key and produces the original plaintext.

The essential steps are the following.

1. Each user generates a pair of keys to be used for the encryption and decryp-
tion of messages.

2. Each user places one of the two keys in a public register or other accessible
file. This is the public key. The companion key is kept private. As Figure 9.1a
suggests, each user maintains a collection of public keys obtained from others.

3. If Bob wishes to send a confidential message to Alice, Bob encrypts the mes-
sage using Alice’s public key.

4. When Alice receives the message, she decrypts it using her private key. No
other recipient can decrypt the message because only Alice knows Alice’s pri-
vate key.

With this approach, all participants have access to public keys, and private
keys are generated locally by each participant and therefore need never be distrib-
uted. As long as a user’s private key remains protected and secret, incoming com-
munication is secure. At any time, a system can change its private key and publish
the companion public key to replace its old public key.

Table 9.2 summarizes some of the important aspects of symmetric and public-
key encryption. To discriminate between the two, we refer to the key used in sym-
metric encryption as a secret key. The two keys used for asymmetric encryption are
referred to as the public key and the private key.2 Invariably, the private key is kept
secret, but it is referred to as a private key rather than a secret key to avoid confu-
sion with symmetric encryption.

Let us take a closer look at the essential elements of a public-key encryption
scheme, using Figure 9.2 (compare with Figure 3.2). There is some source A that
produces a message in plaintext, X = [X1, X2, c , XM]. The M elements of X are
letters in some finite alphabet. The message is intended for destination B. B gener-
ates a related pair of keys: a public key, PUb, and a private key, PRb. PRb is known
only to B, whereas PUb is publicly available and therefore accessible by A.

With the message X and the encryption key PUb as input, A forms the cipher-
text Y = [Y1, Y2, c , YN]:

Y = E(PUb, X)

The intended receiver, in possession of the matching private key, is able to invert
the transformation:

X = D(PRb,Y)

2The following notation is used consistently throughout. A secret key is represented by Km, where m is
some modifier; for example, Ka is a secret key owned by user A. A public key is represented by PUa, for
user A, and the corresponding private key is PRa. Encryption of plaintext X can be performed with a
secret key, a public key, or a private key, denoted by E(Ka, X), E(PUa, X), and E(PRa, X), respectively.
Similarly, decryption of ciphertext Y can be performed with a secret key, a public key, or a private key,
denoted by D(Ka, Y), D(PUa, Y), and D(PRa, Y), respectively.

9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 289

An adversary, observing Y and having access to PUb, but not having access to PRb
or X, must attempt to recover X and/or PRb. It is assumed that the adversary does
have knowledge of the encryption (E) and decryption (D) algorithms. If the ad-
versary is interested only in this particular message, then the focus of effort is to
recover X by generating a plaintext estimate Xn . Often, however, the adversary is
interested in being able to read future messages as well, in which case an attempt is
made to recover PRb by generating an estimate PRnb.

Conventional Encryption Public-Key Encryption

Needed to Work:

1. The same algorithm with the same key is
used for encryption and decryption.

2. The sender and receiver must share the
algorithm and the key.

Needed for Security:

1. The key must be kept secret.

2. It must be impossible or at least impractical
to decipher a message if the key is kept
secret.

3. Knowledge of the algorithm plus samples of
ciphertext must be insufficient to determine
the key.

Needed to Work:

1. One algorithm is used for encryption and a related
algorithm for decryption with a pair of keys, one for
encryption and one for decryption.

2. The sender and receiver must each have one of the
matched pair of keys (not the same one).

Needed for Security:

1. One of the two keys must be kept secret.

2. It must be impossible or at least impractical to
decipher a message if one of the keys is kept secret.

3. Knowledge of the algorithm plus one of the keys
plus samples of ciphertext must be insufficient to
determine the other key.

Table 9.2 Conventional and Public-Key Encryption

Figure 9.2 Public-Key Cryptosystem: Confidentiality

Message
source

Cryptanalyst

Key pair
source

Destination
X

P̂Rb

PUb

Encryption
algorithm

Decryption
algorithm

PRb

X̂

Source A Destination B

Y = E[PUb, X] X =
D[PRb, Y]

Hiva-Network.Com

290 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

Figure 9.3 Public-Key Cryptosystem: Authentication

Message
source

Cryptanalyst

Key pair
source

Destination
X

^

PRa

PRa

PUa

Encryption
algorithm

Decryption
algorithm

Source A Destination B

Y = E[PRa, X] X =
D[PUa, Y]

We mentioned earlier that either of the two related keys can be used for en-
cryption, with the other being used for decryption. This enables a rather differ-
ent cryptographic scheme to be implemented. Whereas the scheme illustrated in
Figure 9.2 provides confidentiality, Figures 9.1b and 9.3 show the use of public-key
encryption to provide authentication:

Y = E(PRa,X)
X = D(PUa,Y)

In this case, A prepares a message to B and encrypts it using A’s private key
before transmitting it. B can decrypt the message using A’s public key. Because the
message was encrypted using A’s private key, only A could have prepared the mes-
sage. Therefore, the entire encrypted message serves as a digital signature. In addi-
tion, it is impossible to alter the message without access to A’s private key, so the
message is authenticated both in terms of source and in terms of data integrity.

In the preceding scheme, the entire message is encrypted, which, although val-
idating both author and contents, requires a great deal of storage. Each document
must be kept in plaintext to be used for practical purposes. A copy also must be
stored in ciphertext so that the origin and contents can be verified in case of a dis-
pute. A more efficient way of achieving the same results is to encrypt a small block
of bits that is a function of the document. Such a block, called an authenticator,
must have the property that it is infeasible to change the document without chang-
ing the authenticator. If the authenticator is encrypted with the sender’s private
key, it serves as a signature that verifies origin, content, and sequencing. Chapter 13
examines this technique in detail.

9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 291

It is important to emphasize that the encryption process depicted in Figures 9.1b
and 9.3 does not provide confidentiality. That is, the message being sent is safe from
alteration but not from eavesdropping. This is obvious in the case of a signature
based on a portion of the message, because the rest of the message is transmitted in
the clear. Even in the case of complete encryption, as shown in Figure 9.3, there is
no protection of confidentiality because any observer can decrypt the message by
using the sender’s public key.

It is, however, possible to provide both the authentication function and confi-
dentiality by a double use of the public-key scheme (Figure 9.4):

Z = E(PUb, E(PRa,X))
X = D(PUa, D(PRb,Z))

In this case, we begin as before by encrypting a message, using the sender’s private
key. This provides the digital signature. Next, we encrypt again, using the receiver’s
public key. The final ciphertext can be decrypted only by the intended receiver, who
alone has the matching private key. Thus, confidentiality is provided. The disadvan-
tage of this approach is that the public-key algorithm, which is complex, must be
exercised four times rather than two in each communication.

Applications for Public-Key Cryptosystems

Before proceeding, we need to clarify one aspect of public-key cryptosystems that
is otherwise likely to lead to confusion. Public-key systems are characterized by the
use of a cryptographic algorithm with two keys, one held private and one available
publicly. Depending on the application, the sender uses either the sender’s private
key or the receiver’s public key, or both, to perform some type of cryptographic

Figure 9.4 Public-Key Cryptosystem: Authentication and Secrecy

Message
source

Message
dest.

X Encryption
algorithm

Key pair
source

PUb PRb

Source A Destination B

Key pair
source

PRa PUa

Y Encryption
algorithm

Z Decryption
algorithm

Y Decryption
algorithm

X

292 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

function. In broad terms, we can classify the use of public-key cryptosystems into
three categories

■ Encryption/decryption: The sender encrypts a message with the recipient’s
public key, and the recipient decrypts the message with the recipient’s private
key.

■ Digital signature: The sender “signs” a message with its private key. Signing
is achieved by a cryptographic algorithm applied to the message or to a small
block of data that is a function of the message.

■ Key exchange: Two sides cooperate to exchange a session key, which is a secret
key for symmetric encryption generated for use for a particular transaction (or
session) and valid for a short period of time. Several different approaches are
possible, involving the private key(s) of one or both parties; this is discussed in
Chapter 10.

Some algorithms are suitable for all three applications, whereas others can be
used only for one or two of these applications. Table 9.3 indicates the applications
supported by the algorithms discussed in this book.

Requirements for Public-Key Cryptography

The cryptosystem illustrated in Figures 9.2 through 9.4 depends on a cryptographic
algorithm based on two related keys. Diffie and Hellman postulated this system
without demonstrating that such algorithms exist. However, they did lay out the
conditions that such algorithms must fulfill [DIFF76b].

1. It is computationally easy for a party B to generate a key pair (public key PUb,
private key PRb).

2. It is computationally easy for a sender A, knowing the public key and the mes-
sage to be encrypted, M, to generate the corresponding ciphertext:

C = E(PUb, M)

3. It is computationally easy for the receiver B to decrypt the resulting ciphertext
using the private key to recover the original message:

M = D(PRb, C) = D[PRb, E(PUb, M)]

4. It is computationally infeasible for an adversary, knowing the public key, PUb,
to determine the private key, PRb.

Algorithm Encryption/Decryption Digital Signature Key Exchange

RSA Yes Yes Yes

Elliptic Curve Yes Yes Yes

Diffie–Hellman No No Yes

DSS No Yes No

Table 9.3 Applications for Public-Key Cryptosystems

9.1 / PRINCIPLES OF PUBLIC-KEY CRYPTOSYSTEMS 293

5. It is computationally infeasible for an adversary, knowing the public key, PUb,
and a ciphertext, C, to recover the original message, M.

We can add a sixth requirement that, although useful, is not necessary for all
public-key applications:

6. The two keys can be applied in either order:

M = D[PUb, E(PRb, M)] = D[PRb, E(PUb, M)]

These are formidable requirements, as evidenced by the fact that only a few
algorithms (RSA, elliptic curve cryptography, Diffie–Hellman, DSS) have received
widespread acceptance in the several decades since the concept of public-key cryp-
tography was proposed.

Before elaborating on why the requirements are so formidable, let us first re-
cast them. The requirements boil down to the need for a trap-door one-way func-
tion. A one-way function3 is one that maps a domain into a range such that every
function value has a unique inverse, with the condition that the calculation of the
function is easy, whereas the calculation of the inverse is infeasible:

Y = f(X) easy
X = f-1(Y) infeasible

Generally, easy is defined to mean a problem that can be solved in polynomial
time as a function of input length. Thus, if the length of the input is n bits, then the
time to compute the function is proportional to na, where a is a fixed constant. Such
algorithms are said to belong to the class P. The term infeasible is a much fuzzier
concept. In general, we can say a problem is infeasible if the effort to solve it grows
faster than polynomial time as a function of input size. For example, if the length
of the input is n bits and the time to compute the function is proportional to 2n,
the problem is considered infeasible. Unfortunately, it is difficult to determine if a
particular algorithm exhibits this complexity. Furthermore, traditional notions of
computational complexity focus on the worst-case or average-case complexity of
an algorithm. These measures are inadequate for cryptography, which requires that
it be infeasible to invert a function for virtually all inputs, not for the worst case or
even average case. A brief introduction to some of these concepts is provided in
Appendix W.

We now turn to the definition of a trap-door one-way function, which is easy
to calculate in one direction and infeasible to calculate in the other direction un-
less certain additional information is known. With the additional information the
inverse can be calculated in polynomial time. We can summarize as follows: A trap-
door one-way function is a family of invertible functions fk, such that

Y = fk(X) easy, if k and X are known
X = fk-1(Y) easy, if k and Y are known
X = fk-1(Y) infeasible, if Y is known but k is not known

3Not to be confused with a one-way hash function, which takes an arbitrarily large data field as its
argument and maps it to a fixed output. Such functions are used for authentication (see Chapter 11).

294 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

Thus, the development of a practical public-key scheme depends on discovery of a
suitable trap-door one-way function.

Public-Key Cryptanalysis

As with symmetric encryption, a public-key encryption scheme is vulnerable to a
brute-force attack. The countermeasure is the same: Use large keys. However, there
is a tradeoff to be considered. Public-key systems depend on the use of some sort of
invertible mathematical function. The complexity of calculating these functions may
not scale linearly with the number of bits in the key but grow more rapidly than that.
Thus, the key size must be large enough to make brute-force attack impractical but
small enough for practical encryption and decryption. In practice, the key sizes that
have been proposed do make brute-force attack impractical but result in encryp-
tion/decryption speeds that are too slow for general-purpose use. Instead, as was
mentioned earlier, public-key encryption is currently confined to key management
and signature applications.

Another form of attack is to find some way to compute the private key given
the public key. To date, it has not been mathematically proven that this form of at-
tack is infeasible for a particular public-key algorithm. Thus, any given algorithm,
including the widely used RSA algorithm, is suspect. The history of cryptanalysis
shows that a problem that seems insoluble from one perspective can be found to
have a solution if looked at in an entirely different way.

Finally, there is a form of attack that is peculiar to public-key systems. This is,
in essence, a probable-message attack. Suppose, for example, that a message were
to be sent that consisted solely of a 56-bit DES key. An adversary could encrypt all
possible 56-bit DES keys using the public key and could discover the encrypted key
by matching the transmitted ciphertext. Thus, no matter how large the key size of the
public-key scheme, the attack is reduced to a brute-force attack on a 56-bit key. This
attack can be thwarted by appending some random bits to such simple messages.

9.2 THE RSA ALGORITHM

The pioneering paper by Diffie and Hellman [DIFF76b] introduced a new approach
to cryptography and, in effect, challenged cryptologists to come up with a crypto-
graphic algorithm that met the requirements for public-key systems. A number of
algorithms have been proposed for public-key cryptography. Some of these, though
initially promising, turned out to be breakable.4

One of the first successful responses to the challenge was developed in 1977
by Ron Rivest, Adi Shamir, and Len Adleman at MIT and first published in 1978
[RIVE78].5 The Rivest-Shamir-Adleman (RSA) scheme has since that time reigned
supreme as the most widely accepted and implemented general-purpose approach
to public-key encryption.

4The most famous of the fallen contenders is the trapdoor knapsack proposed by Ralph Merkle. We
describe this in Appendix J.
5Apparently, the first workable public-key system for encryption/decryption was put forward by Clifford
Cocks of Britain’s CESG in 1973 [COCK73]; Cocks’ method is virtually identical to RSA.

9.2 / THE RSA ALGORITHM 295

The RSA scheme is a cipher in which the plaintext and ciphertext are integers
between 0 and n - 1 for some n. A typical size for n is 1024 bits, or 309 decimal
digits. That is, n is less than 21024. We examine RSA in this section in some detail,
beginning with an explanation of the algorithm. Then we examine some of the com-
putational and cryptanalytical implications of RSA.

Description of the Algorithm

RSA makes use of an expression with exponentials. Plaintext is encrypted in blocks,
with each block having a binary value less than some number n. That is, the block
size must be less than or equal to log2(n) + 1; in practice, the block size is i bits,
where 2i 6 n … 2i+ 1. Encryption and decryption are of the following form, for
some plaintext block M and ciphertext block C.

C = Me mod n
M = Cd mod n = (Me)d mod n = Med mod n

Both sender and receiver must know the value of n. The sender knows
the value of e, and only the receiver knows the value of d. Thus, this is a public-
key encryption algorithm with a public key of PU = {e, n} and a private key of
PR = {d, n}. For this algorithm to be satisfactory for public-key encryption, the fol-
lowing requirements must be met.

1. It is possible to find values of e, d, and n such that Med mod n = M for all M 6 n.
2. It is relatively easy to calculate Me mod n and Cd mod n for all values of M 6 n.
3. It is infeasible to determine d given e and n.

For now, we focus on the first requirement and consider the other questions
later. We need to find a relationship of the form

Med mod n = M

The preceding relationship holds if e and d are multiplicative inverses modulo f(n),
where f(n) is the Euler totient function. It is shown in Chapter 2 that for p, q prime,
f(pq) = (p - 1)(q - 1). The relationship between e and d can be expressed as

ed mod f(n) = 1 (9.1)

This is equivalent to saying

ed K 1 mod f(n)
d K e-1 mod f(n)

That is, e and d are multiplicative inverses mod f(n). Note that, according to the
rules of modular arithmetic, this is true only if d (and therefore e) is relatively
prime to f(n). Equivalently, gcd(f(n), d) = 1. See Appendix R for a proof that
Equation (9.1) satisfies the requirement for RSA.

We are now ready to state the RSA scheme. The ingredients are the following:
p, q, two prime numbers (private, chosen)
n = pq (public, calculated)
e, with gcd(f(n), e) = 1; 1 6 e 6 f(n) (public, chosen)
d K e-1 (mod f(n)) (private, calculated)

296 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

The private key consists of {d, n} and the public key consists of {e, n}. Suppose
that user A has published its public key and that user B wishes to send the message
M to A. Then B calculates C = Me mod n and transmits C. On receipt of this ci-
phertext, user A decrypts by calculating M = Cd mod n.

Figure 9.5 summarizes the RSA algorithm. It corresponds to Figure 9.1a: Alice
generates a public/private key pair; Bob encrypts using Alice’s public key; and Alice
decrypts using her private key. An example from [SING99] is shown in Figure 9.6.
For this example, the keys were generated as follows.

1. Select two prime numbers, p = 17 and q = 11.
2. Calculate n = pq = 17 * 11 = 187.
3. Calculate f(n) = (p - 1)(q - 1) = 16 * 10 = 160.
4. Select e such that e is relatively prime to f(n) = 160 and less than f(n); we

choose e = 7.
5. Determine d such that de K 1 (mod 160) and d 6 160. The correct value is

d = 23, because 23 * 7 = 161 = (1 * 160) + 1; d can be calculated using
the extended Euclid’s algorithm (Chapter 2).

The resulting keys are public key PU = {7, 187} and private key PR = {23, 187}.
The example shows the use of these keys for a plaintext input of M = 88. For
encryption, we need to calculate C = 887 mod 187. Exploiting the properties of
modular arithmetic, we can do this as follows.

887 mod 187 = [(884 mod 187) * (882 mod 187)
* (881 mod 187)] mod 187
881 mod 187 = 88

882 mod 187 = 7744 mod 187 = 77

884 mod 187 = 59,969,536 mod 187 = 132

887 mod 187 = (88 * 77 * 132) mod 187 = 894,432 mod 187 = 11

For decryption, we calculate M = 1123 mod 187:

1123 mod 187 = [(111 mod 187) * (112 mod 187) * (114 mod 187)
* (118 mod 187) * (118 mod 187)] mod 187
111 mod 187 = 11

112 mod 187 = 121

114 mod 187 = 14,641 mod 187 = 55

118 mod 187 = 214,358,881 mod 187 = 33

1123 mod 187 = (11 * 121 * 55 * 33 * 33) mod 187
= 79,720,245 mod 187 = 88

We now look at an example from [HELL79], which shows the use of RSA to
process multiple blocks of data. In this simple example, the plaintext is an alpha-
numeric string. Each plaintext symbol is assigned a unique code of two decimal

9.2 / THE RSA ALGORITHM 297

digits (e.g., a = 00, A = 26).6 A plaintext block consists of four decimal digits, or
two alphanumeric characters. Figure 9.7a illustrates the sequence of events for the
encryption of multiple blocks, and Figure 9.7b gives a specific example. The circled
numbers indicate the order in which operations are performed.

Computational Aspects

We now turn to the issue of the complexity of the computation required to use
RSA. There are actually two issues to consider: encryption/decryption and key
generation. Let us look first at the process of encryption and decryption and then
consider key generation.

6 The complete mapping of alphanumeric characters to decimal digits is at box.com/Crypto7e in the doc-
ument RSAexample.pdf.

Figure 9.6 Example of RSA Algorithm

Encryption

Plaintext
88

Plaintext
88

Ciphertext
1188 mod 187 = 11

PU = 7, 187

Decryption

7 11 mod 187 = 88

PR � 23, 187

23

Figure 9.5 The RSA Algorithm

Key Generation by Alice

Select p, q p and q both prime, p ≠ q

Calculate n = p * q

Calcuate f(n) = (p - 1)(q - 1)

Select integer e gcd (f(n), e) = 1; 1 6 e 6 f(n)

Calculate d d K e-1 (mod f(n))

Public key PU = {e, n}

Private key PR = {d, n}

Encryption by Bob with Alice’s Public Key

Plaintext: M 6 n

Ciphertext: C = Me mod n

Decryption by Alice with Alice’s Public Key

Ciphertext: C

Plaintext: M = Cd mod n

298 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

EXPONENTIATION IN MODULAR ARITHMETIC Both encryption and decryption in RSA
involve raising an integer to an integer power, mod n. If the exponentiation is done
over the integers and then reduced modulo n, the intermediate values would be
gargantuan. Fortunately, as the preceding example shows, we can make use of a
property of modular arithmetic:

[(a mod n) * (b mod n)] mod n = (a * b) mod n

Thus, we can reduce intermediate results modulo n. This makes the calculation
practical.

Another consideration is the efficiency of exponentiation, because with RSA,
we are dealing with potentially large exponents. To see how efficiency might be in-
creased, consider that we wish to compute x16. A straightforward approach requires
15 multiplications:

x16 = x * x * x * x * x * x * x * x * x * x * x * x * x * x * x * x

Figure 9.7 RSA Processing of Multiple Blocks

Plaintext P

Decimal string

Sender

Receiver

(a) General approach (b) Example

Blocks of numbers

Transmit

P1, P2,

P1 = C1
d mod n

P2 = C2
d mod n

Ciphertext C

C1 = P1
e mod n

C2 = P2
e mod n

Recovered
decimal text

n = pq

Random number
generator

e, p, q

Private key
d, n

Public key
e, n

How_are_you?

33 14 22 62 00 17 04 62 24 14 20 66

Sender

Receiver

Transmit

P1 = 3314 P2 = 2262 P3 = 0017
P4 = 0462 P5 = 2414 P6 = 2066

C1 = 3314
11 mod 11023 = 10260

C2 = 2262
11 mod 11023 = 9489

C3 = 17
11 mod 11023 = 1782

C4 = 462
11 mod 11023 = 727

C5 = 2414
11 mod 11023 = 10032

C6 = 2066
11 mod 11023 = 2253

P1 = 10260
5891 mod 11023 = 3314

P2 = 9489
5891 mod 11023 = 2262

P3 = 1782
5891 mod 11023 = 0017

P4 = 727
5891 mod 11023 = 0462

P5 = 10032
5891 mod 11023 = 2414

P6 = 2253
5891 mod 11023 = 2066

11023 = 73 151

5891 = 11–1 mod 10800
10800 = (73 – 1)(151 – 1)
11023 = 73 51

Random number
generator

e = 11
n = 11023

d = 5891
n = 11023

e = 11
p = 73, q = 151

1

2

6

3

4

5

7

1

2

6

3

4

5

7

d = e–1 mod f(n)
f(n) = (p – 1)(q – 1)

n = pq

Hiva-Network.Com

9.2 / THE RSA ALGORITHM 299

However, we can achieve the same final result with only four multiplications if we
repeatedly take the square of each partial result, successively forming (x2, x4, x8, x16).
As another example, suppose we wish to calculate x11 mod n for some integers x
and n. Observe that x11 = x1 + 2 + 8 = (x)(x2)(x8). In this case, we compute x mod n,
x2 mod n, x4 mod n, and x8 mod n and then calculate [(x mod n) * (x2 mod n) *
(x8 mod n)] mod n.

More generally, suppose we wish to find the value ab mod n with a, b, and m
positive integers. If we express b as a binary number bkbk - 1 c b0, then we have

b = a
bi≠0

2i

Therefore,

ab = a
¢ Σ2i

bi≠0
≤
= q

bi≠0
a(2

i)

ab mod n = J q
bi≠0

a(2
i) R mod n = ¢ q

bi≠0
Ja(2i) mod nR ≤ mod n

We can therefore develop the algorithm7 for computing ab mod n, shown in
Figure 9.8. Table 9.4 shows an example of the execution of this algorithm. Note that
the variable c is not needed; it is included for explanatory purposes. The final value
of c is the value of the exponent.

EFFICIENT OPERATION USING THE PUBLIC KEY To speed up the operation of the
RSA algorithm using the public key, a specific choice of e is usually made. The most
common choice is 65537 (216 + 1); two other popular choices are 3 and 17. Each of
these choices has only two 1 bits, so the number of multiplications required to per-
form exponentiation is minimized.

7The algorithm has a long history; this particular pseudocode expression is from [CORM09].

Figure 9.8 Algorithm for Computing ab mod n

c 0; f 1

c 2 × cdo

bi = 1

then c c + 1

if

f (f × f) mod n

f (f × a) mod n

for i k downto 0

return f

Note: The integer b is expressed as a
binary number bkbk - 1cb0.

300 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

However, with a very small public key, such as e = 3, RSA becomes vulner-
able to a simple attack. Suppose we have three different RSA users who all use
the value e = 3 but have unique values of n, namely (n1, n2, n3). If user A sends
the same encrypted message M to all three users, then the three ciphertexts are
C1 = M3 mod n1, C2 = M3 mod n2, and C3 = M3 mod n3. It is likely that n1, n2,
and n3 are pairwise relatively prime. Therefore, one can use the Chinese remainder
theorem (CRT) to compute M3 mod (n1n2n3). By the rules of the RSA algorithm,
M is less than each of the ni; therefore M

3 6 n1n2n3. Accordingly, the attacker need
only compute the cube root of M3. This attack can be countered by adding a unique
pseudorandom bit string as padding to each instance of M to be encrypted. This ap-
proach is discussed subsequently.

The reader may have noted that the definition of the RSA algorithm
(Figure 9.5) requires that during key generation the user selects a value of e that is
relatively prime to f(n). Thus, if a value of e is selected first and the primes p and q
are generated, it may turn out that gcd(f(n), e) ≠ 1. In that case, the user must
reject the p, q values and generate a new p, q pair.

EFFICIENT OPERATION USING THE PRIVATE KEY We cannot similarly choose a small
constant value of d for efficient operation. A small value of d is vulnerable to a
brute-force attack and to other forms of cryptanalysis [WIEN90]. However, there
is a way to speed up computation using the CRT. We wish to compute the value
M = Cd mod n. Let us define the following intermediate results:

Vp = Cd mod p Vq = Cd mod q

Following the CRT using Equation (8.8), define the quantities

Xp = q * (q-1 mod p) Xq = p * (p-1 mod q)

The CRT then shows, using Equation (8.9), that

M = (VpXp + VqXq) mod n

Furthermore, we can simplify the calculation of Vp and Vq using Fermat’s
theorem, which states that ap - 1 K 1 (mod p) if p and a are relatively prime. Some
thought should convince you that the following are valid.

Vp = Cd mod p = Cd mod(p - 1) mod p Vq = Cd mod q = Cd mod(q - 1) mod q

i 9 8 7 6 5 4 3 2 1 0

bi 1 0 0 0 1 1 0 0 0 0
c 1 2 4 8 17 35 70 140 280 560
f 7 49 157 526 160 241 298 166 67 1

Table 9.4 Result of the Fast Modular Exponentiation Algorithm for ab mod n, where a = 7,
b = 560 = 1000110000, and n = 561

9.2 / THE RSA ALGORITHM 301

The quantities d mod (p - 1) and d mod (q - 1) can be precalculated. The
end result is that the calculation is approximately four times as fast as evaluating
M = Cd mod n directly [BONE02].

KEY GENERATION Before the application of the public-key cryptosystem, each par-
ticipant must generate a pair of keys. This involves the following tasks.

■ Determining two prime numbers, p and q.

■ Selecting either e or d and calculating the other.

First, consider the selection of p and q. Because the value of n = pq will be
known to any potential adversary, in order to prevent the discovery of p and q
by exhaustive methods, these primes must be chosen from a sufficiently large set
(i.e., p and q must be large numbers). On the other hand, the method used for find-
ing large primes must be reasonably efficient.

At present, there are no useful techniques that yield arbitrarily large primes,
so some other means of tackling the problem is needed. The procedure that is gen-
erally used is to pick at random an odd number of the desired order of magnitude
and test whether that number is prime. If not, pick successive random numbers until
one is found that tests prime.

A variety of tests for primality have been developed (e.g., see [KNUT98] for
a description of a number of such tests). Almost invariably, the tests are probabi-
listic. That is, the test will merely determine that a given integer is probably prime.
Despite this lack of certainty, these tests can be run in such a way as to make the
probability as close to 1.0 as desired. As an example, one of the more efficient
and popular algorithms, the Miller–Rabin algorithm, is described in Chapter 2.
With this algorithm and most such algorithms, the procedure for testing whether
a given integer n is prime is to perform some calculation that involves n and a
randomly chosen integer a. If n “fails” the test, then n is not prime. If n “passes”
the test, then n may be prime or nonprime. If n passes many such tests with many
different randomly chosen values for a, then we can have high confidence that n
is, in fact, prime.

In summary, the procedure for picking a prime number is as follows.

1. Pick an odd integer n at random (e.g., using a pseudorandom number
generator).

2. Pick an integer a 6 n at random.
3. Perform the probabilistic primality test, such as Miller–Rabin, with a as a

parameter. If n fails the test, reject the value n and go to step 1.

4. If n has passed a sufficient number of tests, accept n; otherwise, go to step 2.

This is a somewhat tedious procedure. However, remember that this process is per-
formed relatively infrequently: only when a new pair (PU, PR) is needed.

It is worth noting how many numbers are likely to be rejected before a
prime number is found. A result from number theory, known as the prime number
theorem, states that the primes near N are spaced on the average one every

302 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

ln (N) integers. Thus, on average, one would have to test on the order of ln(N)
integers before a prime is found. Actually, because all even integers can be im-
mediately rejected, the correct figure is ln(N)/2. For example, if a prime on the
order of magnitude of 2200 were sought, then about ln(2200)/2 = 70 trials would be
needed to find a prime.

Having determined prime numbers p and q, the process of key generation is
completed by selecting a value of e and calculating d or, alternatively, selecting a
value of d and calculating e. Assuming the former, then we need to select an e such
that gcd(f(n), e) = 1 and then calculate d K e-1 (mod f(n)). Fortunately, there is
a single algorithm that will, at the same time, calculate the greatest common divi-
sor of two integers and, if the gcd is 1, determine the inverse of one of the integers
modulo the other. The algorithm, referred to as the extended Euclid’s algorithm,
is explained in Chapter 2. Thus, the procedure is to generate a series of random
numbers, testing each against f(n) until a number relatively prime to f(n) is found.
Again, we can ask the question: How many random numbers must we test to find
a usable number, that is, a number relatively prime to f(n)? It can be shown easily
that the probability that two random numbers are relatively prime is about 0.6; thus,
very few tests would be needed to find a suitable integer (see Problem 2.18).

The Security of RSA

Five possible approaches to attacking the RSA algorithm are

■ Brute force: This involves trying all possible private keys.

■ Mathematical attacks: There are several approaches, all equivalent in effort to
factoring the product of two primes.

■ Timing attacks: These depend on the running time of the decryption algorithm.

■ Hardware fault-based attack: This involves inducing hardware faults in the
processor that is generating digital signatures.

■ Chosen ciphertext attacks: This type of attack exploits properties of the RSA
algorithm.

The defense against the brute-force approach is the same for RSA as for other
cryptosystems, namely, to use a large key space. Thus, the larger the number of bits
in d, the better. However, because the calculations involved, both in key generation
and in encryption/decryption, are complex, the larger the size of the key, the slower
the system will run.

In this subsection, we provide an overview of mathematical and timing attacks.

THE FACTORING PROBLEM We can identify three approaches to attacking RSA
mathematically.

1. Factor n into its two prime factors. This enables calculation of f(n) =
(p - 1) * (q - 1), which in turn enables determination of d K e-1 (mod f(n)).

2. Determine f(n) directly, without first determining p and q. Again, this enables
determination of d K e-1 (mod f(n)).

3. Determine d directly, without first determining f(n).

9.2 / THE RSA ALGORITHM 303

Most discussions of the cryptanalysis of RSA have focused on the task of
factoring n into its two prime factors. Determining f(n) given n is equivalent to
factoring n [RIBE96]. With presently known algorithms, determining d given
e and n appears to be at least as time-consuming as the factoring problem [KALI95].
Hence, we can use factoring performance as a benchmark against which to evaluate
the security of RSA.

For a large n with large prime factors, factoring is a hard problem, but it is not
as hard as it used to be. A striking illustration of this is the following. In 1977, the
three inventors of RSA dared Scientific American readers to decode a cipher they
printed in Martin Gardner’s “Mathematical Games” column [GARD77]. They of-
fered a $100 reward for the return of a plaintext sentence, an event they predicted
might not occur for some 40 quadrillion years. In April of 1994, a group working
over the Internet claimed the prize after only eight months of work [LEUT94]. This
challenge used a public key size (length of n) of 129 decimal digits, or around 428
bits. In the meantime, just as they had done for DES, RSA Laboratories had issued
challenges for the RSA cipher with key sizes of 100, 110, 120, and so on, digits. The
latest challenge to be met is the RSA-768 challenge with a key length of 232 decimal
digits, or 768 bits. Table 9.5 shows the results.

A striking fact about the progress reflected in Table 9.5 concerns the method
used. Until the mid-1990s, factoring attacks were made using an approach known
as the quadratic sieve. The attack on RSA-130 used a newer algorithm, the gen-
eralized number field sieve (GNFS), and was able to factor a larger number than
RSA-129 at only 20% of the computing effort.

The threat to larger key sizes is twofold: the continuing increase in computing
power and the continuing refinement of factoring algorithms. We have seen that
the move to a different algorithm resulted in a tremendous speedup. We can expect
further refinements in the GNFS, and the use of an even better algorithm is also
a possibility. In fact, a related algorithm, the special number field sieve (SNFS),

Number of Decimal Digits Number of Bits Date Achieved

100 332 April 1991

110 365 April 1992

120 398 June 1993

129 428 April 1994

130 431 April 1996

140 465 February 1999

155 512 August 1999

160 530 April 2003

174 576 December 2003

200 663 May 2005

193 640 November 2005

232 768 December 2009

Table 9.5 Progress in RSA Factorization

304 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

can factor numbers with a specialized form considerably faster than the generalized
number field sieve. Figure 9.9 compares the performance of the two algorithms. It is
reasonable to expect a breakthrough that would enable a general factoring perfor-
mance in about the same time as SNFS, or even better [ODLY95]. Thus, we need
to be careful in choosing a key size for RSA. The team that produced the 768-bit
factorization [KLEI10] observed that factoring a 1024-bit RSA modulus would be
about a thousand times harder than factoring a 768-bit modulus, and a 768-bit RSA
modulus is several thousands times harder to factor than a 512-bit one. Based on the
amount of time between the 512-bit and 768-bit factorization successes, the team
felt it to be reasonable to expect that the 1024-bit RSA moduli could be factored
well within the next decade by a similar academic effort. Thus, they recommended
phasing out usage of 1024-bit RSA within the next few years (from 2010).

Figure 9.9 MIPS-years Needed to Factor

1022

M
IP

S-
ye

ar
s n

ee
de

d
to

fa
ct

or

200018001600140012001000800600
Bits

General number
field sieve

Special number
field sieve

1020

1018

1016

1014

1012

1010

108

106

104

102

100

9.2 / THE RSA ALGORITHM 305

In addition to specifying the size of n, a number of other constraints have been
suggested by researchers. To avoid values of n that may be factored more easily, the
algorithm’s inventors suggest the following constraints on p and q.

1. p and q should differ in length by only a few digits. Thus, for a 1024-bit key
(309 decimal digits), both p and q should be on the order of magnitude of
1075 to 10100.

2. Both (p - 1) and (q - 1) should contain a large prime factor.
3. gcd(p - 1, q - 1) should be small.

In addition, it has been demonstrated that if e 6 n and d 6 n1/4, then d can be easily
determined [WIEN90].

TIMING ATTACKS If one needed yet another lesson about how difficult it is to assess
the security of a cryptographic algorithm, the appearance of timing attacks provides
a stunning one. Paul Kocher, a cryptographic consultant, demonstrated that a
snooper can determine a private key by keeping track of how long a computer takes
to decipher messages [KOCH96, KALI96b]. Timing attacks are applicable not just
to RSA, but to other public-key cryptography systems. This attack is alarming for
two reasons: It comes from a completely unexpected direction, and it is a ciphertext-
only attack.

A timing attack is somewhat analogous to a burglar guessing the combi-
nation of a safe by observing how long it takes for someone to turn the dial
from number to number. We can explain the attack using the modular expo-
nentiation algorithm of Figure 9.8, but the attack can be adapted to work with
any implementation that does not run in fixed time. In this algorithm, modular
exponentiation is accomplished bit by bit, with one modular multiplication per-
formed at each iteration and an additional modular multiplication performed
for each 1 bit.

As Kocher points out in his paper, the attack is simplest to understand in an
extreme case. Suppose the target system uses a modular multiplication function that
is very fast in almost all cases but in a few cases takes much more time than an entire
average modular exponentiation. The attack proceeds bit-by-bit starting with the
leftmost bit, bk. Suppose that the first j bits are known (to obtain the entire exponent,
start with j = 0 and repeat the attack until the entire exponent is known). For a
given ciphertext, the attacker can complete the first j iterations of the for loop. The
operation of the subsequent step depends on the unknown exponent bit. If the bit
is set, d d (d * a) mod n will be executed. For a few values of a and d, the modu-
lar multiplication will be extremely slow, and the attacker knows which these are.
Therefore, if the observed time to execute the decryption algorithm is always slow
when this particular iteration is slow with a 1 bit, then this bit is assumed to be 1.
If a number of observed execution times for the entire algorithm are fast, then this
bit is assumed to be 0.

In practice, modular exponentiation implementations do not have such ex-
treme timing variations, in which the execution time of a single iteration can ex-
ceed the mean execution time of the entire algorithm. Nevertheless, there is enough
variation to make this attack practical. For details, see [KOCH96].

306 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

Although the timing attack is a serious threat, there are simple countermea-
sures that can be used, including the following.

■ Constant exponentiation time: Ensure that all exponentiations take the same
amount of time before returning a result. This is a simple fix but does degrade
performance.

■ Random delay: Better performance could be achieved by adding a random
delay to the exponentiation algorithm to confuse the timing attack. Kocher
points out that if defenders don’t add enough noise, attackers could still suc-
ceed by collecting additional measurements to compensate for the random
delays.

■ Blinding: Multiply the ciphertext by a random number before performing ex-
ponentiation. This process prevents the attacker from knowing what cipher-
text bits are being processed inside the computer and therefore prevents the
bit-by-bit analysis essential to the timing attack.

RSA Data Security incorporates a blinding feature into some of its products.
The private-key operation M = Cd mod n is implemented as follows.

1. Generate a secret random number r between 0 and n - 1.
2. Compute C′ = C(r e) mod n, where e is the public exponent.
3. Compute M′ = (C′)d mod n with the ordinary RSA implementation.
4. Compute M = M′r -1 mod n. In this equation, r -1 is the multiplicative inverse

of r mod n; see Chapter 2 for a discussion of this concept. It can be demon-
strated that this is the correct result by observing that r ed mod n = r mod n.

RSA Data Security reports a 2 to 10% performance penalty for blinding.

FAULT-BASED ATTACK Still another unorthodox approach to attacking RSA is re-
ported in [PELL10]. The approach is an attack on a processor that is generating
RSA digital signatures. The attack induces faults in the signature computation by
reducing the power to the processor. The faults cause the software to produce in-
valid signatures, which can then be analyzed by the attacker to recover the private
key. The authors show how such an analysis can be done and then demonstrate it
by extracting a 1024-bit private RSA key in approximately 100 hours, using a com-
mercially available microprocessor.

The attack algorithm involves inducing single-bit errors and observing the re-
sults. The details are provided in [PELL10], which also references other proposed
hardware fault-based attacks against RSA.

This attack, while worthy of consideration, does not appear to be a serious
threat to RSA. It requires that the attacker have physical access to the target ma-
chine and that the attacker is able to directly control the input power to the pro-
cessor. Controlling the input power would for most hardware require more than
simply controlling the AC power, but would also involve the power supply control
hardware on the chip.

9.2 / THE RSA ALGORITHM 307

CHOSEN CIPHERTEXT ATTACK AND OPTIMAL ASYMMETRIC ENCRYPTION PADDING The
basic RSA algorithm is vulnerable to a chosen ciphertext attack (CCA). CCA is
defined as an attack in which the adversary chooses a number of ciphertexts and
is then given the corresponding plaintexts, decrypted with the target’s private key.
Thus, the adversary could select a plaintext, encrypt it with the target’s public key,
and then be able to get the plaintext back by having it decrypted with the private
key. Clearly, this provides the adversary with no new information. Instead, the ad-
versary exploits properties of RSA and selects blocks of data that, when processed
using the target’s private key, yield information needed for cryptanalysis.

A simple example of a CCA against RSA takes advantage of the following
property of RSA:

E(PU, M1) * E(PU, M2) = E(PU, [M1 * M2]) (9.2)

We can decrypt C = Me mod n using a CCA as follows.

1. Compute X = (C * 2e) mod n.
2. Submit X as a chosen ciphertext and receive back Y = Xd mod n.

But now note that

X = (C mod n) * (2e mod n)
= (Me mod n) * (2e mod n)
= (2M)e mod n

Therefore, Y = (2M) mod n. From this, we can deduce M. To overcome this
simple attack, practical RSA-based cryptosystems randomly pad the plaintext prior
to encryption. This randomizes the ciphertext so that Equation (9.2) no longer
holds. However, more sophisticated CCAs are possible, and a simple padding with a
random value has been shown to be insufficient to provide the desired security. To
counter such attacks, RSA Security Inc., a leading RSA vendor and former holder
of the RSA patent, recommends modifying the plaintext using a procedure known
as optimal asymmetric encryption padding (OAEP). A full discussion of the threats
and OAEP are beyond our scope; see [POIN02] for an introduction and [BELL94]
for a thorough analysis. Here, we simply summarize the OAEP procedure.

Figure 9.10 depicts OAEP encryption. As a first step, the message M to be en-
crypted is padded. A set of optional parameters, P, is passed through a hash func-
tion, H.8 The output is then padded with zeros to get the desired length in the overall
data block (DB). Next, a random seed is generated and passed through another hash
function, called the mask generating function (MGF). The resulting hash value is bit-
by-bit XORed with DB to produce a maskedDB. The maskedDB is in turn passed
through the MGF to form a hash that is XORed with the seed to produce the masked-
seed. The concatenation of the maskedseed and the maskedDB forms the encoded
message EM. Note that the EM includes the padded message, masked by the seed,
and the seed, masked by the maskedDB. The EM is then encrypted using RSA.

8A hash function maps a variable-length data block or message into a fixed-length value called a hash
code. Hash functions are discussed in depth in Chapter 11.

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308 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

9.3 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Figure 9.10 Encryption Using Optimal Asymmetric
Encryption Padding (OAEP)

Seed

Maskedseed

DB

MaskedDB

M

EM

Padding

H(P)

MGF

MGF

P

P = encoding parameters
M = message to be encoded
H = hash function

DB = data block
MGF = mask generating function
EM = encoded message

Key Terms

chosen ciphertext attack
(CCA)

digital signature
key exchange
one-way function

optimal asymmetric encryption
padding (OAEP)

private key
public key
public-key cryptography

public-key cryptosystems
public-key encryption
RSA
timing attack
trap-door one-way function

Review Questions

9.1 What is a public key certificate?
9.2 What are the roles of the public and private key?
9.3 What are three broad categories of applications of public-key cryptosystems?

9.3 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 309

9.4 What requirements must a public-key cryptosystems fulfill to be a secure algorithm?
9.5 How can a probable-message attack be used for public-key cryptanalysis?
9.6 List the different approaches to attack the RSA algorithm.
9.7 Describe the countermeasures to be used against the timing attack.

Problems

9.1 Prior to the discovery of any specific public-key schemes, such as RSA, an existence
proof was developed whose purpose was to demonstrate that public-key encryption is
possible in theory. Consider the functions f1(x1) = z1; f2(x2, y2) = z2; f3(x3, y3) = z3,
where all values are integers with 1 … xi, yi, zi … N. Function f1 can be represented
by a vector M1 of length N, in which the kth entry is the value of f1(k). Similarly, f2
and f3 can be represented by N * N matrices M2 and M3. The intent is to represent
the encryption/decryption process by table lookups for tables with very large values
of N. Such tables would be impractically huge but could be constructed in principle.
The scheme works as follows: Construct M1 with a random permutation of all inte-
gers between 1 and N; that is, each integer appears exactly once in M1. Construct M2
so that each row contains a random permutation of the first N integers. Finally, fill in
M3 to satisfy the following condition:

f3(f2(f1(k), p), k) = p for all k, p with 1 … k, p , … N

To summarize,
1. M1 takes an input k and produces an output x.
2. M2 takes inputs x and p giving output z.
3. M3 takes inputs z and k and produces p.
The three tables, once constructed, are made public.
a. It should be clear that it is possible to construct M3 to satisfy the preceding condi-

tion. As an example, fill in M3 for the following simple case:

4 3 5 2 4 1

3 4 2 5 3 1

M1 = 2 M2 = 5 4 3 1 2 M3 =

5 1 3 2 5 4

1 2 1 4 3 5

Convention: The ith element of M1 corresponds to k = i. The ith row of M2 cor-
responds to x = i; the jth column of M2 corresponds to p = j. The ith row of M3
corresponds to z = i; the jth column of M3 corresponds to k = j.

b. Describe the use of this set of tables to perform encryption and decryption be-
tween two users.

c. Argue that this is a secure scheme.
9.2 Perform encryption and decryption using the RSA algorithm, as in Figure 9.5, for the

following:
a. p = 3 ; q = 7 , e = 5 ; M = 1 0
b. p = 5 ; q = 1 3 , e = 5 ; M = 8
c. p = 7 ; q = 1 7 , e = 1 1 ; M = 1 1
d. p = 7 ; q = 1 3 , e = 1 1 ; M = 2
e. p = 1 7 ; q = 2 3 , e = 9 ; M = 7
Hint: Decryption is not as hard as you think; use some finesse.

9.3 In a public-key system using RSA, you intercept the ciphertext C = 2 0 sent to a user
whose public key is e = 1 3 , n = 7 7 . What is the plaintext M?

310 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

9.4 In an RSA system, the public key of a given user is e = 6 5 , n = 2 8 8 1 . What is the
private key of this user? Hint: First use trial-and-error to determine p and q; then use
the extended Euclidean algorithm to find the multiplicative inverse of 31 modulo
f(n).

9.5 In using the RSA algorithm, if a small number of repeated encodings give back the
plaintext, what is the likely cause?

9.6 Suppose we have a set of blocks encoded with the RSA algorithm and we don’t have
the private key. Assume n = pq, e is the public key. Suppose also someone tells us
they know one of the plaintext blocks has a common factor with n. Does this help us
in any way?

9.7 In the RSA public-key encryption scheme, each user has a public key, e, and a private
key, d. Suppose Bob leaks his private key. Rather than generating a new modulus, he
decides to generate a new public and a new private key. Is this safe?

9.8 Suppose Bob uses the RSA cryptosystem with a very large modulus n for which the
factorization cannot be found in a reasonable amount of time. Suppose Alice sends
a message to Bob by representing each alphabetic character as an integer between
0 and 25 (A S 0, c , Z S 25) and then encrypting each number separately using
RSA with large e and large n. Is this method secure? If not, describe the most effi-
cient attack against this encryption method.

9.9 Using a spreadsheet (such as Excel) or a calculator, perform the operations described
below. Document results of all intermediate modular multiplications. Determine a
number of modular multiplications per each major transformation (such as encryp-
tion, decryption, primality testing, etc.).
a. Test all odd numbers in the range from 215 to 223 for primality using the Miller–

Rabin test with base 2.
b. Encrypt the message block M = 2 using RSA with the following parameters:

e = 23 and n = 233 * 241.
c. Compute a private key (d, p, q) corresponding to the given above public key (e, n).
d. Perform the decryption of the obtained ciphertext

1. without using the Chinese Remainder Theorem, and
2. using the Chinese Remainder Theorem.

9.10 Assume that you generate an authenticated and encrypted message by first applying the
RSA transformation determined by your private key, and then enciphering the mes-
sage using recipient’s public key (note that you do NOT use hash function before the
first transformation). Will this scheme work correctly [i.e., give the possibility to recon-
struct the original message at the recipient’s side, for all possible relations between the
sender’s modulus nS and the recipient’s modulus nR (nS 7 nR, nS 6 nR, nS = nR)]?
Explain your answer. In case your answer is “no,” how would you correct this scheme?

9.11 “I want to tell you, Holmes,” Dr. Watson’s voice was enthusiastic, “that your recent
activities in network security have increased my interest in cryptography. And just
yesterday I found a way to make one-time pad encryption practical.”

“Oh, really?” Holmes’ face lost its sleepy look.
“Yes, Holmes. The idea is quite simple. For a given one-way function F, I gener-

ate a long pseudorandom sequence of elements by applying F to some standard se-
quence of arguments. The cryptanalyst is assumed to know F and the general nature
of the sequence, which may be as simple as S, S + 1, S + 2, c , but not secret S.
And due to the one-way nature of F, no one is able to extract S given F(S + i) for
some i, thus even if he somehow obtains a certain segment of the sequence, he will
not be able to determine the rest.”

“I am afraid, Watson, that your proposal isn’t without flaws and at least it needs
some additional conditions to be satisfied by F. Let’s consider, for instance, the RSA
encryption function, that is F(M) = MK mod N, K is secret. This function is believed
to be one-way, but I wouldn’t recommend its use, for example, on the sequence
M = 2, 3, 4, 5, 6, . . . ”

9.3 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 311

“But why, Holmes?” Dr. Watson apparently didn’t understand. “Why do you
think that the resulting sequence 2K mod N, 3K mod N, 4K mod N, . . . is not appropri-
ate for one-time pad encryption if K is kept secret?”

“Because it is—at least partially—predictable, dear Watson, even if K is kept se-
cret. You have said that the cryptanalyst is assumed to know F and the general nature
of the sequence. Now let’s assume that he will obtain somehow a short segment of the
output sequence. In crypto circles, this assumption is generally considered to be a vi-
able one. And for this output sequence, knowledge of just the first two elements will
allow him to predict quite a lot of the next elements of the sequence, even if not all of
them, thus this sequence can’t be considered to be cryptographically strong. And with
the knowledge of a longer segment he could predict even more of the next elements
of the sequence. Look, knowing the general nature of the sequence and its first two
elements 2K mod N and 3K mod N, you can easily compute its following elements.”

Show how this can be done.
9.12 Show how RSA can be represented by matrices M1, M2, and M3 of Problem 9.1.
9.13 Consider the following scheme:

1. Pick an odd number, E.
2. Pick two prime numbers, P and Q, where (P - 1)(Q - 1) - 1 is evenly divisible

by E.
3. Multiply P and Q to get N.

4. Calculate D =
(P - 1)(Q - 1)(E - 1) + 1

E

Is this scheme equivalent to RSA? Show why or why not.
9.14 Consider the following scheme by which B encrypts a message for A.

1. A chooses two large primes P and Q that are also relatively prime to (P - 1)
and (Q - 1).

2. A publishes N = PQ as its public key.
3. A calculates P= and Q= such that PP= K 1 (mod Q - 1) and QQ= K 1 (mod P - 1).
4. B encrypts message M as C = MN mod N.
5. A finds M by solving M K CP

=
(mod Q) and M K CQ

=
(mod P).

a. Explain how this scheme works.
b. How does it differ from RSA?
c. Is there any particular advantage to RSA compared to this scheme?
d. Show how this scheme can be represented by matrices M1, M2, and M3 of

Problem 9.1.

9.15 “This is a very interesting case, Watson,” Holmes said. “The young man loves a girl,
and she loves him too. However, her father is a strange fellow who insists that his
would-be son-in-law must design a simple and secure protocol for an appropriate
public-key cryptosystem he could use in his company’s computer network. The young
man came up with the following protocol for communication between two parties.
For example, user A wishing to send message M to user B: (messages exchanged are
in the format sender’s name, text, receiver’s name)”
1. A sends B the following block: (A, E(PUb, [M, A]), B).
2. B acknowledges receipt by sending to A the following block: (B, E(PUa, [M, B]), A).
“You can see that the protocol is really simple. But the girl’s father claims that the
young man has not satisfied his call for a simple protocol, because the proposal con-
tains a certain redundancy and can be further simplified to the following:”
1. A sends B the block: (A, E(PUb, M), B).
2. B acknowledges receipt by sending to A the block: (B, E(PUa, M), A).
“On the basis of that, the girl’s father refuses to allow his daughter to marry the
young man, thus making them both unhappy. The young man was just here to ask
me for help.”

312 CHAPTER 9 / PUBLIC-KEY CRYPTOGRAPHY AND RSA

“Hmm, I don’t see how you can help him.” Watson was visibly unhappy with the
idea that the sympathetic young man has to lose his love.

“Well, I think I could help. You know, Watson, redundancy is sometimes good to
ensure the security of protocol. Thus, the simplification the girl’s father has proposed
could make the new protocol vulnerable to an attack the original protocol was able
to resist,” mused Holmes. “Yes, it is so, Watson. Look, all an adversary needs is to
be one of the users of the network and to be able to intercept messages exchanged
between A and B. Being a user of the network, he has his own public encryption key
and is able to send his own messages to A or to B and to receive theirs. With the help
of the simplified protocol, he could then obtain message M user A has previously sent
to B using the following procedure:”

Complete the description.
9.16 Use the fast exponentiation algorithm of Figure 9.8 to determine 6 4 7 2 mod 3415.

Show the steps involved in the computation.
9.17 Here is another realization of the fast exponentiation algorithm. Demonstrate that it

is equivalent to the one in Figure 9.8.
1. f d 1; T d a; E d b
2. if odd(E) then f d f : T
3. E d [ E/2 ]
4. T d T : T
5. if E + 0 then goto 2
6. output f

9.18 This problem illustrates a simple application of the chosen ciphertext attack. Bob
intercepts a ciphertext C intended for Alice and encrypted with Alice’s public key e.
Bob wants to obtain the original message M = Cd mod n. Bob chooses a random
value r less than n and computes

Z = r e mod n
X = ZC mod n
t = r -1 mod n

Next, Bob gets Alice to authenticate (sign) X with her private key (as in Figure 9.3),
thereby decrypting X. Alice returns Y = Xd mod n. Show how Bob can use the infor-
mation now available to him to determine M.

9.19 Show the OAEP decoding operation used for decryption that corresponds to the
encoding operation of Figure 9.10.

9.20 Improve on algorithm P1 in Appendix W.
a. Develop an algorithm that requires 2n multiplications and n + 1 additions. Hint:

xi+ 1 = xi * x.
b. Develop an algorithm that requires only n + 1 multiplications and n + 1 addi-

tions. Hint: P(x) = a0 + x * q(x), where q(x) is a polynomial of degree (n - 1).
Note: The remaining problems concern the knapsack public-key algorithm described
in Appendix J.

9.21 What items are in the knapsack in Figure F.1?
9.22 Perform encryption and decryption using the knapsack algorithm for the following:

a. a= = (1, 5, 7, 14); w = 11; m = 30; x = 1011
b. a= = (1, 2, 7, 12, 23, 38, 116, 248); w = 201; m = 457; x = 10101010
c. a= = (2, 4, 7, 15, 29); w = 36; m = 47; x = 10011
d. a= = (15, 92, 108, 279, 563, 1172, 2243, 4468); w = 2033; m = 8764; x = 10110011

9.23 Why is it a requirement that m 7 a
n

1=1
a= i?

313

Other Public-Key
Cryptosystems

10.1 Diffie–Hellman Key Exchange

The Algorithm
Key Exchange Protocols
Man-in-the-Middle Attack

10.2 Elgamal Cryptographic System

10.3 Elliptic Curve Arithmetic

Abelian Groups
Elliptic Curves over Real Numbers
Elliptic Curves over Zp
Elliptic Curves over GF(2m)

10.4 Elliptic Curve Cryptography

Analog of Diffie–Hellman Key Exchange
Elliptic Curve Encryption/Decryption
Security of Elliptic Curve Cryptography

10.5 Pseudorandom Number Generation Based on an Asymmetric Cipher

PRNG Based on RSA
PRNG Based on Elliptic Curve Cryptography

10.6 Key Terms, Review Questions, and Problems

CHAPTER

314 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

This chapter begins with a description of one of the earliest and simplest PKCS:
Diffie–Hellman key exchange. The chapter then looks at another important scheme,
the Elgamal PKCS. Next, we look at the increasingly important PKCS known as elliptic
curve cryptography. Finally, the use of public-key algorithms for pseudorandom num-
ber generation is examined.

10.1 DIFFIE–HELLMAN KEY EXCHANGE

The first published public-key algorithm appeared in the seminal paper by Diffie
and Hellman that defined public-key cryptography [DIFF76b] and is generally re-
ferred to as Diffie–Hellman key exchange.1 A number of commercial products em-
ploy this key exchange technique.

The purpose of the algorithm is to enable two users to securely exchange a
key that can then be used for subsequent symmetric encryption of messages. The
algorithm itself is limited to the exchange of secret values.

The Diffie–Hellman algorithm depends for its effectiveness on the difficulty
of computing discrete logarithms. Briefly, we can define the discrete logarithm in
the following way. Recall from Chapter 2 that a primitive root of a prime number p
is one whose powers modulo p generate all the integers from 1 to p - 1. That is, if
a is a primitive root of the prime number p, then the numbers

a mod p, a2 mod p, c , ap - 1 mod p

are distinct and consist of the integers from 1 through p - 1 in some permutation.
For any integer b and a primitive root a of prime number p, we can find a

unique exponent i such that

b K ai (mod p) where 0 … i … (p - 1)

1Williamson of Britain’s CESG published the identical scheme a few months earlier in a classified docu-
ment [WILL76] and claims to have discovered it several years prior to that; see [ELLI99] for a discussion.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Define Diffie–Hellman key exchange.

◆ Understand the man-in-the-middle attack.

◆ Present an overview of the Elgamal cryptographic system.

◆ Understand elliptic curve arithmetic.

◆ Present an overview of elliptic curve cryptography.

◆ Present two techniques for generating pseudorandom numbers using an
asymmetric cipher.

10.1 / DIFFIE–HELLMAN KEY EXCHANGE 315

The exponent i is referred to as the discrete logarithm of b for the base a, mod p. We
express this value as dloga,p(b). See Chapter 2 for an extended discussion of discrete
logarithms.

The Algorithm

Figure 10.1 summarizes the Diffie–Hellman key exchange algorithm. For this
scheme, there are two publicly known numbers: a prime number q and an inte-
ger a that is a primitive root of q. Suppose the users A and B wish to create a
shared key.

User A selects a random integer XA 6 q and computes YA = aXA mod q.
Similarly, user B independently selects a random integer XB 6 q and computes
YB = aXB mod q. Each side keeps the X value private and makes the Y value avail-
able publicly to the other side. Thus, XA is A’s private key and YA is A’s correspond-
ing public key, and similarly for B. User A computes the key as K = (YB)XA mod q
and user B computes the key as K = (YA)XB mod q. These two calculations produce
identical results:

Figure 10.1 The Diffie–Hellman Key Exchange

Alice Bob

Alice and Bob share a
prime number q and an
integer A, such that A < q and
A is a primitive root of q

Alice generates a private
key XA such that XA < q

Alice calculates a public
key YA = AXA mod q

Alice receives Bob’s
public key YB in plaintext

Alice calculates shared
secret key K = (YB)XA mod q

Bob calculates shared
secret key K = (YA)XB mod q

Bob receives Alice’s
public key YA in plaintext

Bob calculates a public
key YB = AXB mod q

Bob generates a private
key XB such that XB < q

Alice and Bob share a
prime number q and an
integer A, such that A < q and
A is a primitive root of q

YA YB

316 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

K = (YB)XA mod q
= (aXB mod q)XA mod q
= (aXB)XA mod q by the rules of modular arithmetic
= aXBXA mod q
= (aXA)XB mod q
= (aXA mod q)XB mod q
= (YA)XB mod q

The result is that the two sides have exchanged a secret value. Typically, this
secret value is used as shared symmetric secret key. Now consider an adversary who
can observe the key exchange and wishes to determine the secret key K. Because
XA and XB are private, an adversary only has the following ingredients to work with:
q, a, YA, and YB. Thus, the adversary is forced to take a discrete logarithm to deter-
mine the key. For example, to determine the private key of user B, an adversary
must compute

XB = dloga,q(YB)

The adversary can then calculate the key K in the same manner as user B calculates
it. That is, the adversary can calculate K as

K = (YA)XB mod q

The security of the Diffie–Hellman key exchange lies in the fact that, while
it is relatively easy to calculate exponentials modulo a prime, it is very difficult
to calculate discrete logarithms. For large primes, the latter task is considered
infeasible.

Here is an example. Key exchange is based on the use of the prime number
q = 353 and a primitive root of 353, in this case a = 3. A and B select private keys
XA = 97 and XB = 233, respectively. Each computes its public key:

A computes YA = 397 mod 353 = 40.
B computes YB = 3233 mod 353 = 248.

After they exchange public keys, each can compute the common secret key:

A computes K = (YB)XA mod 353 = 24897 mod 353 = 160.
B computes K = (YA)XB mod 353 = 40233 mod 353 = 160.

We assume an attacker would have available the following information:

q = 353; a = 3; YA = 40; YB = 248

In this simple example, it would be possible by brute force to determine the secret
key 160. In particular, an attacker E can determine the common key by discovering
a solution to the equation 3a mod 353 = 40 or the equation 3b mod 353 = 248. The
brute-force approach is to calculate powers of 3 modulo 353, stopping when the re-
sult equals either 40 or 248. The desired answer is reached with the exponent value
of 97, which provides 397 mod 353 = 40.

With larger numbers, the problem becomes impractical.

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10.1 / DIFFIE–HELLMAN KEY EXCHANGE 317

Key Exchange Protocols

Figure 10.1 shows a simple protocol that makes use of the Diffie–Hellman calcula-
tion. Suppose that user A wishes to set up a connection with user B and use a secret
key to encrypt messages on that connection. User A can generate a one-time pri-
vate key XA, calculate YA, and send that to user B. User B responds by generating
a private value XB, calculating YB, and sending YB to user A. Both users can now
calculate the key. The necessary public values q and a would need to be known
ahead of time. Alternatively, user A could pick values for q and a and include those
in the first message.

As an example of another use of the Diffie–Hellman algorithm, suppose that a
group of users (e.g., all users on a LAN) each generate a long-lasting private value Xi
(for user i) and calculate a public value Yi. These public values, together with global
public values for q and a, are stored in some central directory. At any time, user j
can access user i’s public value, calculate a secret key, and use that to send an en-
crypted message to user A. If the central directory is trusted, then this form of com-
munication provides both confidentiality and a degree of authentication. Because
only i and j can determine the key, no other user can read the message (confidential-
ity). Recipient i knows that only user j could have created a message using this key
(authentication). However, the technique does not protect against replay attacks.

Man-in-the-Middle Attack

The protocol depicted in Figure 10.1 is insecure against a man-in-the-middle attack.
Suppose Alice and Bob wish to exchange keys, and Darth is the adversary. The at-
tack proceeds as follows (Figure 10.2).

1. Darth prepares for the attack by generating two random private keys XD1 and
XD2 and then computing the corresponding public keys YD1 and YD2.

2. Alice transmits YA to Bob.

3. Darth intercepts YA and transmits YD1 to Bob. Darth also calculates
K2 = (YA)XD2 mod q.

4. Bob receives YD1 and calculates K1 = (YD1)XB mod q.
5. Bob transmits YB to Alice.

6. Darth intercepts YB and transmits YD2 to Alice. Darth calculates
K1 = (YB)XD1 mod q.

7. Alice receives YD2 and calculates K2 = (YD2)XA mod q.

At this point, Bob and Alice think that they share a secret key, but instead
Bob and Darth share secret key K1 and Alice and Darth share secret key K2. All
future communication between Bob and Alice is compromised in the following way.

1. Alice sends an encrypted message M: E(K2, M).

2. Darth intercepts the encrypted message and decrypts it to recover M.

3. Darth sends Bob E(K1, M) or E(K1, M=), where M= is any message. In the first
case, Darth simply wants to eavesdrop on the communication without altering
it. In the second case, Darth wants to modify the message going to Bob.

318 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

The key exchange protocol is vulnerable to such an attack because it does not
authenticate the participants. This vulnerability can be overcome with the use of digital
signatures and public-key certificates; these topics are explored in Chapters 13 and 14.

10.2 ELGAMAL CRYPTOGRAPHIC SYSTEM

In 1984, T. Elgamal announced a public-key scheme based on discrete logarithms,
closely related to the Diffie–Hellman technique [ELGA84, ELGA85]. The Elgamal2
cryptosystem is used in some form in a number of standards including the digital
signature standard (DSS), which is covered in Chapter 13, and the S/MIME email
standard (Chapter 19).

2For no apparent reason, most of the literature uses the term ElGamal, although Mr. Elgamal’s last name
does not have a capital letter G.

Figure 10.2 Man-in-the-Middle Attack

Alice Darth Bob

Private key XA
Public key
YA = AXA mod q

Private key XB
Public key
YB = AXB mod q

Private keys XD1, XD2
Public keys
YD1 = AXD1 mod q
YD2 = AXD2 mod q

YA

Secret key
K2 = (YA)XD2 mod q

Secret key
K1 = (YB)XD1 mod q

Secret key
K1 = (YD1)XB mod q

Secret key
K2 = (YD2)XA mod q

Alice and Darth
share K2

Bob and Darth
share K1

YD2 YD1

YB

10.2 / ELGAMAL CRYPTOGRAPHIC SYSTEM 319

As with Diffie–Hellman, the global elements of Elgamal are a prime number q and
a, which is a primitive root of q. User A generates a private/public key pair as follows:

1. Generate a random integer XA, such that 1 6 XA 6 q - 1.
2. Compute YA = aXA mod q.
3. A’s private key is XA and A’s public key is {q, a, YA}.

Any user B that has access to A’s public key can encrypt a message as follows:

1. Represent the message as an integer M in the range 0 … M … q - 1. Longer
messages are sent as a sequence of blocks, with each block being an integer
less than q.

2. Choose a random integer k such that 1 … k … q - 1.
3. Compute a one-time key K = (YA)k mod q.
4. Encrypt M as the pair of integers (C1, C2) where

C1 = ak mod q; C2 = KM mod q

User A recovers the plaintext as follows:

1. Recover the key by computing K = (C1)XA mod q.
2. Compute M = (C2K-1) mod q.

These steps are summarized in Figure 10.3. It corresponds to Figure 9.1a:
Alice generates a public/private key pair; Bob encrypts using Alice’s public key; and
Alice decrypts using her private key.

Let us demonstrate why the Elgamal scheme works. First, we show how K is
recovered by the decryption process:

K = (YA)k mod q K is defined during the encryption process
K = (aXA mod q)k mod q substitute using YA = aXA mod q
K = akXA mod q by the rules of modular arithmetic
K = (C1)XA mod q substitute using C1 = ak mod q

Next, using K, we recover the plaintext as

C2 = KM mod q
(C2K

-1) mod q = KMK-1 mod q = M mod q = M

We can restate the Elgamal process as follows, using Figure 10.3.

1. Bob generates a random integer k.

2. Bob generates a one-time key K using Alice’s public-key components YA, q,
and k.

3. Bob encrypts k using the public-key component a, yielding C1. C1 provides
sufficient information for Alice to recover K.

4. Bob encrypts the plaintext message M using K.

5. Alice recovers K from C1 using her private key.

6. Alice uses K-1 to recover the plaintext message from C2.

320 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

Thus, K functions as a one-time key, used to encrypt and decrypt the message.
For example, let us start with the prime field GF(19); that is, q = 19. It has

primitive roots {2, 3, 10, 13, 14, 15}, as shown in Table 2.7. We choose a = 10.
Alice generates a key pair as follows:

1. Alice chooses XA = 5.
2. Then YA = aXA mod q = a5 mod 19 = 3 (see Table 2.7).
3. Alice’s private key is 5 and Alice’s public key is {q, a, YA} = {19, 10, 3}.

Suppose Bob wants to send the message with the value M = 17. Then:

Figure 10.3 The Elgamal Cryptosystem

Global Public Elements

q prime number

a a 6 q and a a primitive root of q

Key Generation by Alice

Select private XA XA 6 q - 1

Calculate YA YA = aXA mod q

Public key {q, a, YA}

Private key XA

Encryption by Bob with Alice’s Public Key

Plaintext: M 6 q

Select random integer k k 6 q

Calculate K K = (YA)k mod q

Calculate C1 C1 = ak mod q

Calculate C2 C2 = KM mod q

Ciphertext: (C1, C2)

Decryption by Alice with Alice’s Private Key

Ciphertext: (C1, C2)

Calculate K K = (C1)XA mod q

Plaintext: M = (C2K-1) mod q

10.3 / ELLIPTIC CURVE ARITHMETIC 321

1. Bob chooses k = 6.
2. Then K = (YA)k mod q = 36 mod 19 = 729 mod 19 = 7.
3. So

C1 = ak mod q = a6 mod 19 = 11
C2 = KM mod q = 7 * 17 mod 19 = 119 mod 19 = 5

4. Bob sends the ciphertext (11, 5).

For decryption:

1. Alice calculates K = (C1)XA mod q = 115 mod 19 = 161051 mod 19 = 7.
2. Then K-1 in GF(19) is 7-1 mod 19 = 11.
3. Finally, M = (C2K-1) mod q = 5 * 11 mod 19 = 55 mod 19 = 17.

If a message must be broken up into blocks and sent as a sequence of encrypted
blocks, a unique value of k should be used for each block. If k is used for more than
one block, knowledge of one block M1 of the message enables the user to compute
other blocks as follows. Let

C1,1 = ak mod q; C2,1 = KM1 mod q
C1,2 = ak mod q; C2,2 = KM2 mod q

Then,

C2,1
C2,2

=
KM1 mod q
KM2 mod q

=
M1 mod q
M2 mod q

If M1 is known, then M2 is easily computed as

M2 = (C2,1)-1 C2,2 M1 mod q

The security of Elgamal is based on the difficulty of computing discrete
logarithms. To recover A’s private key, an adversary would have to compute
XA = dloga,q(YA). Alternatively, to recover the one-time key K, an adversary
would have to determine the random number k, and this would require computing
the discrete logarithm k = dloga,q(C1). [STIN06] points out that these calculations
are regarded as infeasible if p is at least 300 decimal digits and q - 1 has at least one
“large” prime factor.

10.3 ELLIPTIC CURVE ARITHMETIC

Most of the products and standards that use public-key cryptography for encryp-
tion and digital signatures use RSA. As we have seen, the key length for secure
RSA use has increased over recent years, and this has put a heavier processing
load on applications using RSA. This burden has ramifications, especially for elec-
tronic commerce sites that conduct large numbers of secure transactions. A com-
peting system challenges RSA: elliptic curve cryptography (ECC). ECC is showing
up in standardization efforts, including the IEEE P1363 Standard for Public-Key
Cryptography.

322 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

The principal attraction of ECC, compared to RSA, is that it appears to offer
equal security for a far smaller key size, thereby reducing processing overhead.

ECC is fundamentally more difficult to explain than either RSA or Diffie–
Hellman, and a full mathematical description is beyond the scope of this book. This
section and the next give some background on elliptic curves and ECC. We begin
with a brief review of the concept of abelian group. Next, we examine the concept
of elliptic curves defined over the real numbers. This is followed by a look at el-
liptic curves defined over finite fields. Finally, we are able to examine elliptic curve
ciphers.

The reader may wish to review the material on finite fields in Chapter 5 before
proceeding.

Abelian Groups

Recall from Chapter 5 that an abelian group G, sometimes denoted by {G, # }, is a
set of elements with a binary operation, denoted by # , that associates to each or-
dered pair (a, b) of elements in G an element (a # b) in G, such that the following
axioms are obeyed:3

(A1) Closure: If a and b belong to G, then a # b is also in G.
(A2) Associative: a # (b # c) = (a # b) # c for all a, b, c in G.
(A3) Identity element: There is an element e in G such that a # e = e # a = a

for all a in G.

(A4) Inverse element: For each a in G there is an element a′ in G such that
a # a′ = a′ # a = e.

(A5) Commutative: a # b = b # a for all a, b in G.
A number of public-key ciphers are based on the use of an abelian group.

For example, Diffie–Hellman key exchange involves multiplying pairs of nonzero
integers modulo a prime number q. Keys are generated by exponentiation over
the group, with exponentiation defined as repeated multiplication. For example,
ak mod q = (a * a * c * a) mod q. To attack Diffie–Hellman, the attacker must

k times
determine k given a and ak; this is the discrete logarithm problem.

For elliptic curve cryptography, an operation over elliptic curves, called addi-
tion, is used. Multiplication is defined by repeated addition. For example,

a * k = (a + a + c + a)

k times
where the addition is performed over an elliptic curve. Cryptanalysis involves deter-
mining k given a and (a * k).

3The operator # is generic and can refer to addition, multiplication, or some other mathematical
operation.

v

v

10.3 / ELLIPTIC CURVE ARITHMETIC 323

An elliptic curve is defined by an equation in two variables with coefficients.
For cryptography, the variables and coefficients are restricted to elements in a finite
field, which results in the definition of a finite abelian group. Before looking at this,
we first look at elliptic curves in which the variables and coefficients are real num-
bers. This case is perhaps easier to visualize.

Elliptic Curves over Real Numbers

Elliptic curves are not ellipses. They are so named because they are described by
cubic equations, similar to those used for calculating the circumference of an ellipse.
In general, cubic equations for elliptic curves take the following form, known as a
Weierstrass equation:

y2 + axy + by = x3 + cx2 + dx + e

where a, b, c, d, e are real numbers and x and y take on values in the real numbers.4
For our purpose, it is sufficient to limit ourselves to equations of the form

y2 = x3 + ax + b (10.1)

Such equations are said to be cubic, or of degree 3, because the highest ex-
ponent they contain is a 3. Also included in the definition of an elliptic curve is a
single element denoted O and called the point at infinity or the zero point, which we
discuss subsequently. To plot such a curve, we need to compute

y = 2x3 + ax + b

For given values of a and b, the plot consists of positive and negative values of y for
each value of x. Thus, each curve is symmetric about y = 0. Figure 10.4 shows two
examples of elliptic curves. As you can see, the formula sometimes produces weird-
looking curves.

Now, consider the set of points E(a, b) consisting of all of the points (x, y) that
satisfy Equation (10.1) together with the element O. Using a different value of the
pair (a, b) results in a different set E(a, b). Using this terminology, the two curves in
Figure 10.4 depict the sets E(-1, 0) and E(1, 1), respectively.

GEOMETRIC DESCRIPTION OF ADDITION It can be shown that a group can be defined
based on the set E(a, b) for specific values of a and b in Equation (10.1), provided
the following condition is met:

4a3 + 27b2 ≠ 0 (10.2)

To define the group, we must define an operation, called addition and denoted by
+ , for the set E(a, b), where a and b satisfy Equation (10.2). In geometric terms, the
rules for addition can be stated as follows: If three points on an elliptic curve lie on a
straight line, their sum is O. From this definition, we can define the rules of addition
over an elliptic curve.

4Note that x and y are true variables, which take on values. This is in contrast to our discussion of polyno-
mial rings and fields in Chapter 5, where was treated as an indeterminate.

324 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

1. O serves as the additive identity. Thus O = -O; for any point P on the elliptic
curve, P + O = P. In what follows, we assume P ≠ O and Q ≠ O.

2. The negative of a point P is the point with the same x coordinate but the nega-
tive of the y coordinate; that is, if P = (x, y), then -P = (x, -y). Note that these
two points can be joined by a vertical line. Note that P + (-P) = P - P = O.

3. To add two points P and Q with different x coordinates, draw a straight line
between them and find the third point of intersection R. It is easily seen that
there is a unique point R that is the point of intersection (unless the line is
tangent to the curve at either P or Q, in which case we take R = P or R = Q,
respectively). To form a group structure, we need to define addition on these
three points: P + Q = -R. That is, we define P + Q to be the mirror image

Figure 10.4 Example of Elliptic Curves

-4

-2

0

2

4

-4

-2

0

2

4

543210-1-2

543210-1-2

(a) y2 = x3 - x

(b) y2 = x3 + x + 1

P

P

Q

Q

-(P + Q)

-(P + Q)

(P + Q)

(P + Q)

10.3 / ELLIPTIC CURVE ARITHMETIC 325

(with respect to the x axis) of the third point of intersection. Figure 10.4 illus-
trates this construction.

4. The geometric interpretation of the preceding item also applies to two points,
P and -P, with the same x coordinate. The points are joined by a vertical line,
which can be viewed as also intersecting the curve at the infinity point. We
therefore have P + (-P) = O, which is consistent with item (2).

5. To double a point Q, draw the tangent line and find the other point of intersec-
tion S. Then Q + Q = 2Q = -S.

With the preceding list of rules, it can be shown that the set E(a, b) is an abe-
lian group.

ALGEBRAIC DESCRIPTION OF ADDITION In this subsection, we present some results
that enable calculation of additions over elliptic curves.5 For two distinct points,
P = (xP, yP) and Q = (xQ, yQ), that are not negatives of each other, the slope of the
line l that joins them is ∆ = (yQ - yP)/(xQ - xP). There is exactly one other point
where l intersects the elliptic curve, and that is the negative of the sum of P and Q.
After some algebraic manipulation, we can express the sum R = P + Q as

xR = ∆2 - xP - xQ
yR = -yP + ∆(xP - xR)

(10.3)

We also need to be able to add a point to itself: P + P = 2P = R. When
yP ≠ 0, the expressions are

xR = ¢ 3xP2 + a2yP ≤2 - 2xP
yR = ¢ 3xP2 + a2yP ≤(xP - xR) - yP (10.4)

Elliptic Curves over Zp
Elliptic curve cryptography makes use of elliptic curves in which the variables and
coefficients are all restricted to elements of a finite field. Two families of elliptic
curves are used in cryptographic applications: prime curves over Zp and binary
curves over GF(2m). For a prime curve over Zp, we use a cubic equation in which
the variables and coefficients all take on values in the set of integers from 0 through
p - 1 and in which calculations are performed modulo p. For a binary curve de-
fined over GF(2m), the variables and coefficients all take on values in GF(2m) and
in calculations are performed over GF(2m). [FERN99] points out that prime curves
are best for software applications, because the extended bit-fiddling operations
needed by binary curves are not required; and that binary curves are best for hard-
ware applications, where it takes remarkably few logic gates to create a powerful,
fast cryptosystem. We examine these two families in this section and the next.

5For derivations of these results, see [KOBL94] or other mathematical treatments of elliptic curves.

Hiva-Network.Com

326 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

There is no obvious geometric interpretation of elliptic curve arithmetic over
finite fields. The algebraic interpretation used for elliptic curve arithmetic over real
numbers does readily carry over, and this is the approach we take.

For elliptic curves over Zp, as with real numbers, we limit ourselves to equa-
tions of the form of Equation (10.1), but in this case with coefficients and variables
limited to Zp:

y2 mod p = (x3 + ax + b) mod p (10.5)

For example, Equation (10.5) is satisfied for a = 1, b = 1, x = 9, y = 7, p = 23:

72 mod 23 = (93 + 9 + 1) mod 23
49 mod 23 = 739 mod 23

3 = 3

Now consider the set Ep(a, b) consisting of all pairs of integers (x, y) that sat-
isfy Equation (10.5), together with a point at infinity O. The coefficients a and b and
the variables x and y are all elements of Zp.

For example, let p = 23 and consider the elliptic curve y2 = x3 + x + 1. In
this case, a = b = 1. Note that this equation is the same as that of Figure 10.4b. The
figure shows a continuous curve with all of the real points that satisfy the equation.
For the set E23(1, 1), we are only interested in the nonnegative integers in the quad-
rant from (0, 0) through (p - 1, p - 1) that satisfy the equation mod p. Table 10.1
lists the points (other than O) that are part of E23(1, 1). Figure 10.5 plots the points
of E23(1, 1); note that the points, with one exception, are symmetric about y = 11.5.

It can be shown that a finite abelian group can be defined based on the set
Ep(a, b) provided that (x

3 + ax + b) mod p has no repeated factors. This is equiva-
lent to the condition

(4a3 + 27b2) mod p ≠ 0 mod p (10.6)

Note that Equation (10.6) has the same form as Equation (10.2).
The rules for addition over Ep(a, b), correspond to the algebraic technique de-

scribed for elliptic curves defined over real numbers. For all points P, Q∈ Ep(a, b):

(0, 1) (6, 4) (12, 19)

(0, 22) (6, 19) (13, 7)

(1, 7) (7, 11) (13, 16)

(1, 16) (7, 12) (17, 3)

(3, 10) (9, 7) (17, 20)

(3, 13) (9, 16) (18, 3)

(4, 0) (11, 3) (18, 20)

(5, 4) (11, 20) (19, 5)

(5, 19) (12, 4) (19, 18)

Table 10.1 Points (other than O) on the
Elliptic Curve E23(1, 1)

10.3 / ELLIPTIC CURVE ARITHMETIC 327

1. P + O = P.
2. If P = (xP, yP), then P + (xP, -yP) = O. The point (xP, -yP) is the nega-

tive of P, denoted as -P. For example, in E23(1, 1), for P = (13, 7), we have
-P = (13, -7). But -7 mod 23 = 16. Therefore, -P = (13, 16), which is also
in E23(1, 1).

3. If P = (xp, yp) and Q = (xQ, yQ) with P ≠ -Q, then R = P + Q = (xR, yR)
is determined by the following rules:

xR = (l2 - xP - xQ) mod p
yR = (l(xP - xR) - yP) mod p

where

l = e ayQ - yPxQ - xP b mod p if P ≠ Q
a3xP

2 + a

2yP
b mod p if P = Q

Figure 10.5 The Elliptic Curve E23(1, 1)

0
0
1
2
3
4
5
6
7
8
9

10
11
12
13
14
15
16
17
18
19
20
21
22

1 2 3 4 5 6 7 8 9 10 11
x

y

12 13 14 15 16 17 18 19 20 21 22

328 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

4. Multiplication is defined as repeated addition; for example, 4P =
P + P + P + P.

For example, let P = (3, 10) and Q = (9, 7) in E23(1, 1). Then

l = a7 - 10
9 - 3

b mod 23 = a -3
6
b mod 23 = a -1

2
b mod 23 = 11

xR = (112 - 3 - 9) mod 23 = 109 mod 23 = 17

yR = (11(3 - 17) - 10) mod 23 = -164 mod 23 = 20

So P + Q = (17, 20). To find 2P,

l = ¢ 3(32) + 1
2 * 10

≤ mod 23 = a 5
20
b mod 23 = a1

4
b mod 23 = 6

The last step in the preceding equation involves taking the multiplicative in-
verse of 4 in Z23. This can be done using the extended Euclidean algorithm defined
in Section 4.4. To confirm, note that (6 * 4) mod 23 = 24 mod 23 = 1.

xR = (62 - 3 - 3) mod 23 = 30 mod 23 = 7
yR = (6(3 - 7) - 10) mod 23 = (-34) mod 23 = 12

and 2P = (7, 12).
For determining the security of various elliptic curve ciphers, it is of some in-

terest to know the number of points in a finite abelian group defined over an elliptic
curve. In the case of the finite group EP(a, b), the number of points N is bounded by

p + 1 - 22p … N … p + 1 + 22p

Note that the number of points in Ep(a, b) is approximately equal to the number of
elements in Zp, namely p elements.

Elliptic Curves over GF(2m)

Recall from Chapter 5 that a finite field GF(2m) consists of 2m elements, together
with addition and multiplication operations that can be defined over polynomials.
For elliptic curves over GF(2m), we use a cubic equation in which the variables and
coefficients all take on values in GF(2m) for some number m and in which calcula-
tions are performed using the rules of arithmetic in GF(2m).

(0, 1) (g5, g3) (g9, g13)

(1, g6) (g5, g11) (g10, g)

(1, g13) (g6, g8) (g10, g8)

(g3, g8) (g6, g14) (g12, 0)

(g3, g13) (g9, g10) (g12, g12)

Table 10.2 Points (other than O) on the
Elliptic Curve E24(g

4, 1)

10.3 / ELLIPTIC CURVE ARITHMETIC 329

It turns out that the form of cubic equation appropriate for cryptographic
applications for elliptic curves is somewhat different for GF(2m) than for Zp. The
form is

y2 + xy = x3 + ax2 + b (10.7)

where it is understood that the variables x and y and the coefficients a and b are ele-
ments of GF(2m) and that calculations are performed in GF(2m).

Now consider the set E2m(a, b) consisting of all pairs of integers (x, y) that sat-
isfy Equation (10.7), together with a point at infinity O.

For example, let us use the finite field GF(24) with the irreducible polynomial
f(x) = x4 + x + 1. This yields a generator g that satisfies f(g) = 0 with a value of
g4 = g + 1, or in binary, g = 0010. We can develop the powers of g as follows.

g0 = 0001 g4 = 0011 g8 = 0101 g12 = 1111

g1 = 0010 g5 = 0110 g9 = 1010 g13 = 1101

g2 = 0100 g6 = 1100 g10 = 0111 g14 = 1001

g3 = 1000 g7 = 1011 g11 = 1110 g15 = 0001

For example, g5 = (g4)(g) = (g + 1)(g) = g2 + g = 0110.
Now consider the elliptic curve y2 + xy = x3 + g4x2 + 1. In this case, a = g4

and b = g0 = 1. One point that satisfies this equation is (g5, g3):

(g3)2 + (g5)(g3) = (g5)3 + (g4)(g5)2 + 1
g6 + g8 = g15 + g14 + 1
1100 + 0101 = 0001 + 1001 + 0001
1001 = 1001

Table 10.2 lists the points (other than O) that are part of E24(g
4, 1). Figure 10.6 plots

the points of E24(g
4, 1).

It can be shown that a finite abelian group can be defined based on the set
E2m(a, b), provided that b ≠ 0. The rules for addition can be stated as follows. For
all points P, Q∈ E2m(a, b):

1. P + O = P.
2. If P = (xP, yP), then P + (xP, xP + yP) = O. The point (xP, xP + yP) is the

negative of P, which is denoted as -P.
3. If P = (xP, yP) and Q = (xQ, yQ) with P ≠ -Q and P ≠ Q, then

R = P + Q = (xR, yR) is determined by the following rules:

xR = l2 + l + xP + xQ + a
yR = l(xP + xR) + xR + yP

where

l =
yQ + yP
xQ + xP

330 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

4. If P = (xP, yP) then R = 2P = (xR, yR) is determined by the following rules:

xR = l2 + l + a

yR = xP2 + (l + 1)xR
where

l = xP +
yP
xP

10.4 ELLIPTIC CURVE CRYPTOGRAPHY

The addition operation in ECC is the counterpart of modular multiplication in
RSA, and multiple addition is the counterpart of modular exponentiation. To form
a cryptographic system using elliptic curves, we need to find a “hard problem” cor-
responding to factoring the product of two primes or taking the discrete logarithm.

Consider the equation Q = kP where Q, P∈ EP(a, b) and k 6 p. It is rela-
tively easy to calculate Q given k and P, but it is hard to determine k given Q and P.
This is called the discrete logarithm problem for elliptic curves.

We give an example taken from the Certicom Web site (www.certicom.
com). Consider the group E23(9,17). This is the group defined by the equation
y2 mod 23 = (x3 + 9x + 17) mod 23. What is the discrete logarithm k of Q = (4, 5)
to the base P = (16, 5)? The brute-force method is to compute multiples of P until
Q is found. Thus,

P = (16,5); 2P = (20, 20); 3P = (14, 14); 4P = (19, 20); 5P = (13, 10);
6P = (7, 3); 7P = (8, 7); 8P = (12, 17); 9P = (4, 5)

Figure 10.6 The Elliptic Curve E24(g
4, 1)

1
1
g

g2
g3
g4
g5
g6
g7
g8
g9

g10
g11
g12
g13
g14

0

g g2 g3 g4 g5 g6 g7 g8 g9 g10 g11

x

y

g12 g13 g14 0

10.4 / ELLIPTIC CURVE CRYPTOGRAPHY 331

Because 9P = (4, 5) = Q, the discrete logarithm Q = (4, 5) to the base
P = (16, 5) is k = 9. In a real application, k would be so large as to make the brute-
force approach infeasible.

In the remainder of this section, we show two approaches to ECC that give the
flavor of this technique.

Analog of Diffie–Hellman Key Exchange

Key exchange using elliptic curves can be done in the following manner. First pick
a large integer q, which is either a prime number p or an integer of the form 2m,
and elliptic curve parameters a and b for Equation (10.5) or Equation (10.7). This
defines the elliptic group of points Eq(a, b). Next, pick a base point G = (x1, y1) in
Ep(a, b) whose order is a very large value n. The order n of a point G on an elliptic
curve is the smallest positive integer n such that nG = 0 and G are parameters of
the cryptosystem known to all participants.

A key exchange between users A and B can be accomplished as follows
(Figure 10.7).

1. A selects an integer nA less than n. This is A’s private key. A then generates a
public key PA = nA * G; the public key is a point in Eq(a, b).

2. B similarly selects a private key nB and computes a public key PB.

3. A generates the secret key k = nA * PB. B generates the secret key
k = nB * PA.

The two calculations in step 3 produce the same result because

nA * PB = nA * (nB * G) = nB * (nA * G) = nB * PA

To break this scheme, an attacker would need to be able to compute k given G
and kG, which is assumed to be hard.

As an example,6 take p = 211; Ep(0, -4), which is equivalent to the curve
y2 = x3 - 4; and G = (2, 2). One can calculate that 240G = O. A’s private key
is nA = 121, so A’s public key is PA = 121(2, 2) = (115, 48). B’s private key is
nB = 203, so B’s public key is 203(2, 3) = (130, 203). The shared secret key is
121(130, 203) = 203(115, 48) = (161, 69).

Note that the secret key is a pair of numbers. If this key is to be used as a ses-
sion key for conventional encryption, then a single number must be generated. We
could simply use the x coordinates or some simple function of the x coordinate.

Elliptic Curve Encryption/Decryption

Several approaches to encryption/decryption using elliptic curves have been ana-
lyzed in the literature. In this subsection, we look at perhaps the simplest. The
first task in this system is to encode the plaintext message m to be sent as an (x, y)
point Pm.

6Provided by Ed Schaefer of Santa Clara University.

332 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

It is the point Pm that will be encrypted as a ciphertext and subsequently decrypted.
Note that we cannot simply encode the message as the x or y coordinate of a point,
because not all such coordinates are in Eq(a, b); for example, see Table 10.1. Again,
there are several approaches to this encoding, which we will not address here, but
suffice it to say that there are relatively straightforward techniques that can be
used.

As with the key exchange system, an encryption/decryption system requires a
point G and an elliptic group Eq(a, b) as parameters. Each user A selects a private
key nA and generates a public key PA = nA * G.

To encrypt and send a message Pm to B, A chooses a random positive integer
k and produces the ciphertext Cm consisting of the pair of points:

Cm = {kG, Pm + kPB}

Note that A has used B’s public key PB. To decrypt the ciphertext, B multiplies the
first point in the pair by B’s private key and subtracts the result from the second
point:

Pm + kPB - nB(kG) = Pm + k(nBG) - nB(kG) = Pm

Figure 10.7 ECC Diffie–Hellman Key Exchange

Global Public Elements

Eq(a, b) elliptic curve with parameters a, b, and q, where q is a
prime or an integer of the form 2m

G point on elliptic curve whose order is large value n

User A Key Generation

Select private nA nA 6 n

Calculate public PA PA = nA * G

User B Key Generation

Select private nB nB 6 n

Calculate public PB PB = nB * G

Calculation of Secret Key by User A

K = nA * PB

Calculation of Secret Key by User B

K = nB * PA

10.4 / ELLIPTIC CURVE CRYPTOGRAPHY 333

A has masked the message Pm by adding kPB to it. Nobody but A knows
the value of k, so even though Pb is a public key, nobody can remove the mask
kPB. However, A also includes a “clue,” which is enough to remove the mask if
one knows the private key nB. For an attacker to recover the message, the attacker
would have to compute k given G and kG, which is assumed to be hard.

Let us consider a simple example. The global public elements are q = 257;
Eq(a, b) = E257(0, -4), which is equivalent to the curve y2 = x3 - 4; and G =
(2, 2). Bob’s private key is nB = 101, and his public key is PB = nBG = 101(2, 2) =
(197, 167). Alice wishes to send a message to Bob that is encoded in the elliptic
point Pm = (112, 26). Alice chooses random integer k = 41 and computes kG =
41(2, 2) = (136, 128), kPB = 41(197, 167) = (68, 84) and Pm + kPB = (112, 26)
+ (68, 84) = (246, 174). Alice sends the ciphertext Cm = (C1, C2) = {(136, 128),
(246, 174)} to Bob. Bob receives the ciphertext and computes C2 - nBC1 =
(246, 174) - 101(136, 128) = (246, 174) - (68, 84) = (112, 26).

Security of Elliptic Curve Cryptography

The security of ECC depends on how difficult it is to determine k given kP and P.
This is referred to as the elliptic curve logarithm problem. The fastest known tech-
nique for taking the elliptic curve logarithm is known as the Pollard rho method.
Table 10.3, from NIST SP 800-57 (Recommendation for Key Management—Part 1:
General, September 2015), compares various algorithms by showing comparable
key sizes in terms of computational effort for cryptanalysis. As can be seen, a con-
siderably smaller key size can be used for ECC compared to RSA.

Based on this analysis, SP 800-57 recommends that at least through 2030, ac-
ceptable key lengths are from 3072 to 14,360 bits for RSA and 256 to 512 bits for
ECC. Similarly, the European Union Agency for Network and Information Security
(ENISA) recommends in their 2014 report (Algorithms, Key Size and Parameters
report—2014, November 2014) minimum key lengths for future system of 3072 bits
and 256 bits for RSA and ECC, respectively.

Symmetric Key
Algorithms

Diffie–Hellman, Digital
Signature Algorithm

RSA
(size of n in bits)

ECC
(modulus size in bits)

80
L = 1024
N = 160 1024 160–223

112
L = 2048
N = 224 2048 224–255

128
L = 3072
N = 256 3072 256–383

192
L = 7680
N = 384 7680 384–511

256
L = 15,360
N = 512 15,360 512+

Note: L = size of public key, N = size of private key.

Table 10.3 Comparable Key Sizes in Terms of Computational
Effort for Cryptanalysis (NIST SP-800-57)

334 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

Analysis indicates that for equal key lengths, the computational effort re-
quired for ECC and RSA is comparable [JURI97]. Thus, there is a computational
advantage to using ECC with a shorter key length than a comparably secure RSA.

10.5 PSEUDORANDOM NUMBER GENERATION BASED
ON AN ASYMMETRIC CIPHER

We noted in Chapter 8 that because a symmetric block cipher produces an appar-
ently random output, it can serve as the basis of a pseudorandom number generator
(PRNG). Similarly, an asymmetric encryption algorithm produces apparently ran-
dom output and can be used to build a PRNG. Because asymmetric algorithms are
typically much slower than symmetric algorithms, asymmetric algorithms are not
used to generate open-ended PRNG bit streams. Rather, the asymmetric approach
is useful for creating a pseudorandom function (PRF) for generating a short pseu-
dorandom bit sequence.

In this section, we examine two PRNG designs based on pseudorandom
functions.

PRNG Based on RSA

For a sufficient key length, the RSA algorithm is considered secure and is a good
candidate to form the basis of a PRNG. Such a PRNG, known as the Micali–Schnorr
PRNG [MICA91], is recommended in the ANSI standard X9.82 (Random Number
Generation) and in the ISO standard 18031 (Random Bit Generation).

The PRNG is illustrated in Figure 10.8. As can be seen, this PRNG has much
the same structure as the output feedback (OFB) mode used as a PRNG (see Figure
8.4b and the portion of Figure 7.6a enclosed with a dashed box). In this case, the
encryption algorithm is RSA rather than a symmetric block cipher. Also, a portion
of the output is fed back to the next iteration of the encryption algorithm and the
remainder of the output is used as pseudorandom bits. The motivation for this sepa-
ration of the output into two distinct parts is so that the pseudorandom bits from
one stage do not provide input to the next stage. This separation should contribute
to forward unpredictability.

Figure 10.8 Micali–Schnorr Pseudorandom Bit Generator

Seed = x0

x1 = r most
significant bits

z1 = k least
significant bits

y1 = x0 mod n
e

n, e, r, k n, e, r, k n, e, r, k

x2 = r most
significant bits

z2 = k least
significant bits

x3 = r most
significant bits

z3 = k least
significant bits

y2 = x1 mod n
e y3 = x2 mod n

e
Encrypt Encrypt Encrypt

Hiva-Network.Com

10.5 / PSEUDORANDOM NUMBER GENERATION BASED ON AN ASYMMETRIC CIPHER 335

We can define the PRNG as follows.

Setup Select p, q primes; n = pq; f(n) = (p - 1)(q - 1). Select e such
that gcd(e, f(n)) = 1. These are the standard RSA setup selections
(see Figure 9.5). In addition, let N = [log2n] + 1 (the bitlength of n).
Select r, k such that r + k = N.

Seed Select a random seed x0 of bitlength r.

Generate Generate a pseudorandom sequence of length k * m using the loop
for i from 1 to m do

yi = xi- 1e mod n
xi = r most significant bits of yi
zi = k least significant bits of yi

Output The output sequence is z1 }z2 } c }zm.

The parameters n, r, e, and k are selected to satisfy the following six
requirements.

1. n = pq n is chosen as the product of two primes to
have the cryptographic strength required of
RSA.

2. 1 6 e 6 f(n); gcd (e, f(n)) = 1 Ensures that the mapping s S se mod n is
1 to 1.

3. re Ú 2N Ensures that the exponentiation requires a
full modular reduction.

4. r Ú 2 * strength Protects against a cryptographic attacks.
5. k, r are multiples of 8 An implementation convenience.

6. k Ú 8; r + k = N All bits are used.

The variable strength in requirement 4 is defined in NIST SP 800-90 as fol-
lows: A number associated with the amount of work (that is, the number of opera-
tions) required to break a cryptographic algorithm or system; a security strength
is specified in bits and is a specific value from the set (112, 128, 192, 256) for this
Recommendation. The amount of work needed is 2strength.

There is clearly a tradeoff between r and k. Because RSA is computation-
ally intensive compared to a block cipher, we would like to generate as many
pseudorandom bits per iteration as possible and therefore would like a large
value of k. However, for cryptographic strength, we would like r to be as large as
possible.

For example, if e = 3 and N = 1024, then we have the inequality 3r 7 1024,
yielding a minimum required size for r of 683 bits. For r set to that size, k = 341
bits are generated for each exponentiation (each RSA encryption). In this case,
each exponentiation requires only one modular squaring of a 683-bit number and
one modular multiplication. That is, we need only calculate (xi * (xi2 mod n))
mod n.

336 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

PRNG Based on Elliptic Curve Cryptography

In this subsection, we briefly summarize a technique developed by the U.S. National
Security Agency (NSA) known as dual elliptic curve PRNG (DEC PRNG). This
technique is recommended in NIST SP 800-90, the ANSI standard X9.82, and the
ISO standard 18031. There has been some controversy regarding both the security
and efficiency of this algorithm compared to other alternatives (e.g., see [SCHO06],
[BROW07]).

[SCHO06] summarizes the algorithm as follows: Let P and Q be two known
points on a given elliptic curve. The seed of the DEC PRNG is a random integer
s0 ∈ {0, 1, c , #E(GF(p)) - 1}, where # E(GF(p)) denotes the number of points
on the curve. Let x denote a function that gives the x-coordinate of a point of the
curve. Let lsbi(s) denote the i least significant bits of an integer s. The DEC PRNG
transforms the seed into the pseudorandom sequence of length 240k, k 7 0, as
follows.

for i = 1 to k do
Set si d x(Si-1 P)
Set ri d lsb240 (x(si Q))

end for
Return r1,...,rk

Given the security concerns expressed for this PRNG, the only motivation for
its use would be that it is used in a system that already implements ECC but does
not implement any other symmetric, asymmetric, or hash cryptographic algorithm
that could be used to build a PRNG.

10.6 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

abelian group
binary curve
cubic equation
Diffie–Hellman key exchange
discrete logarithm

elliptic curve
elliptic curve arithmetic
elliptic curve cryptography
finite field
man-in-the-middle attack

Micali–Schnorr
prime curve
primitive root
zero point

Review Questions

10.1 Briefly explain Diffie–Hellman key exchange.
10.2 What is an elliptic curve?
10.3 What is the zero point of an elliptic curve?
10.4 What is the sum of three points on an elliptic curve that lie on a straight line?

10.6 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 337

Problems

10.1 Alice and Bob use the Diffie–Hellman key exchange technique with a common prime
q = 1 5 7 and a primitive root a = 5.
a. If Alice has a private key XA = 15, find her public key YA.
b. If Bob has a private key XB = 27, find his public key YB.
c. What is the shared secret key between Alice and Bob?

10.2 Alice and Bob use the Diffie-Hellman key exchange technique with a common prime
q = 2 3 and a primitive root a = 5 .
a. If Bob has a public key YB = 1 0 , what is Bob’s private key YB?
b. If Alice has a public key YA = 8 , what is the shared key K with Bob?
c. Show that 5 is a primitive root of 23.

10.3 In the Diffie–Hellman protocol, each participant selects a secret number x and sends
the other participant ax mod q for some public number a. What would happen if the
participants sent each other xa for some public number a instead? Give at least one
method Alice and Bob could use to agree on a key. Can Eve break your system with-
out finding the secret numbers? Can Eve find the secret numbers?

10.4 This problem illustrates the point that the Diffie–Hellman protocol is not secure
without the step where you take the modulus; i.e. the “Indiscrete Log Problem” is
not a hard problem! You are Eve and have captured Alice and Bob and imprisoned
them. You overhear the following dialog.

Bob: Oh, let’s not bother with the prime in the Diffie–Hellman protocol, it will
make things easier.

Alice: Okay, but we still need a base a to raise things to. How about a = 3?

Bob: All right, then my result is 27.

Alice: And mine is 243.

What is Bob’s private key XB and Alice’s private key XA? What is their secret com-
bined key? (Don’t forget to show your work.)

10.5 Section 10.1 describes a man-in-the-middle attack on the Diffie–Hellman key
exchange protocol in which the adversary generates two public–private key pairs for
the attack. Could the same attack be accomplished with one pair? Explain.

10.6 Suppose Alice and Bob use an Elgamal scheme with a common prime q = 1 5 7 and
a primitive root a = 5 .
a. If Bob has public key YB = 1 0 and Alice chose the random integer k = 3 , what

is the ciphertext of M = 9 ?
b. If Alice now chooses a different value of k so that the encoding of M = 9 is

C = (2 5 , C2), what is the integer C2?
10.7 Rule (5) for doing arithmetic in elliptic curves over real numbers states that to double

a point Q2, draw the tangent line and find the other point of intersection S. Then
Q + Q = 2Q = -S. If the tangent line is not vertical, there will be exactly one point
of intersection. However, suppose the tangent line is vertical? In that case, what is the
value 2Q? What is the value 3Q?

10.8 Demonstrate that the two elliptic curves of Figure 10.4 each satisfy the conditions for
a group over the real numbers.

10.9 Is (5, 12) a point on the elliptic curve y2 = x 3 + 4 x - 1 over real numbers?

10.10 On the elliptic curve over the real numbers y2 = x3 -
17
12

x + 1, Let P = (0,1) and
Q = (1.5,1.5). Find P + Q and 2P.

10.11 Does the elliptic curve equation y2 = x 3 + x + 2 define a group over Z7?

338 CHAPTER 10 / OTHER PUBLIC-KEY CRYPTOSYSTEMS

10.12 Consider the elliptic curve E7(2,1); that is, the curve is defined by y
2 = x 3 + 2 x + 1

with a modulus of p = 7 . Determine all of the points in E7(2, 1). Hint: Start by calcu-
lating the right-hand side of the equation for all values of x.

10.13 What are the negatives of the following elliptic curve points over Z7? P = (3, 5);
Q = (2, 5); R = (5, 0).

10.14 For E11(1, 7), consider the point G = (3, 2). Compute the multiple of G from 2G
through 13G.

10.15 This problem performs elliptic curve encryption/decryption using the scheme out-
lined in Section 10.4. The cryptosystem parameters are E11(1, 7) and G = (3, 2). B’s
private key is nB = 7.
a. Find B’s public key PB.
b. A wishes to encrypt the message Pm = (10, 7) and chooses the random value

k = 5. Determine the ciphertext Cm.
c. Show the calculation by which B recovers Pm from Cm.

10.16 The following is a first attempt at an elliptic curve signature scheme. We have a global
elliptic curve, prime p, and “generator” G. Alice picks a private signing key XA and
forms the public verifying key YA = XAG. To sign a message M:

■ Alice picks a value k.
■ Alice sends Bob M, k, and the signature S = M - kXAG.
■ Bob verifies that M = S + kYA.

a. Show that this scheme works. That is, show that the verification process produces
an equality if the signature is valid.

b. Show that the scheme is unacceptable by describing a simple technique for forging
a user’s signature on an arbitrary message.

10.17 Here is an improved version of the scheme given in the previous problem. As before,
we have a global elliptic curve, prime p, and “generator” G. Alice picks a private
signing key XA and forms the public verifying key YA = XAG. To sign a message M:

■ Bob picks a value k.
■ Bob sends Alice C1 = kG.
■ Alice sends Bob M and the signature S = M - XAC1.
■ Bob verifies that M = S + kYA.

a. Show that this scheme works. That is, show that the verification process produces
an equality if the signature is valid.

b. Show that forging a message in this scheme is as hard as breaking (Elgamal)
elliptic curve cryptography. (Or find an easier way to forge a message?)

c. This scheme has an extra “pass” compared to other cryptosystems and signature
schemes we have looked at. What are some drawbacks to this?

PART FOUR: CRYPTOGRAPHIC DATA
INTEGRITY ALGORITHMS

CHAPTER

Cryptographic Hash Functions
11.1 Applications of Cryptographic Hash Functions

Message Authentication
Digital Signatures
Other Applications

11.2 Two Simple Hash Functions

11.3 Requirements and Security

Security Requirements for Cryptographic Hash Functions
Brute-Force Attacks
Cryptanalysis

11.4 Hash Functions Based on Cipher Block Chaining

11.5 Secure Hash Algorithm (SHA)

SHA-512 Logic
SHA-512 Round Function
Example

11.6 SHA-3

The Sponge Construction
The SHA-3 Iteration Function f

11.7 Key Terms, Review Questions, and Problems

339

340 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

A hash function H accepts a variable-length block of data M as input and produces
a fixed-size hash value h = H(M). A “good” hash function has the property that the
results of applying the function to a large set of inputs will produce outputs that are
evenly distributed and apparently random. In general terms, the principal object of
a hash function is data integrity. A change to any bit or bits in M results, with high
probability, in a change to the hash value.

The kind of hash function needed for security applications is referred to as a
cryptographic hash function. A cryptographic hash function is an algorithm for which
it is computationally infeasible (because no attack is significantly more efficient than
brute force) to find either (a) a data object that maps to a pre-specified hash result
(the one-way property) or (b) two data objects that map to the same hash result (the
collision-free property). Because of these characteristics, hash functions are often used
to determine whether or not data has changed.

Figure 11.1 depicts the general operation of a cryptographic hash function.
Typically, the input is padded out to an integer multiple of some fixed length
(e.g., 1024 bits), and the padding includes the value of the length of the original mes-
sage in bits. The length field is a security measure to increase the difficulty for an
attacker to produce an alternative message with the same hash value, as explained
subsequently.

This chapter begins with a discussion of the wide variety of applications for
cryptographic hash functions. Next, we look at the security requirements for such
functions. Then we look at the use of cipher block chaining to implement a crypto-
graphic hash function. The remainder of the chapter is devoted to the most important
and widely used family of cryptographic hash functions, the Secure Hash Algorithm
(SHA) family.

Appendix N describes MD5, a well-known cryptographic hash function with
similarities to SHA-1.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Summarize the applications of cryptographic hash functions.

◆ Explain why a hash function used for message authentication needs to be
secured.

◆ Understand the differences among preimage resistant, second preimage
resistant, and collision resistant properties.

◆ Present an overview of the basic structure of cryptographic hash functions.

◆ Describe how cipher block chaining can be used to construct a hash function.

◆ Understand the operation of SHA-512.

◆ Understand the birthday paradox and present an overview of the birthday
attack.

11.1 / APPLICATIONS OF CRYPTOGRAPHIC HASH FUNCTIONS 341

11.1 APPLICATIONS OF CRYPTOGRAPHIC HASH FUNCTIONS

Perhaps the most versatile cryptographic algorithm is the cryptographic hash func-
tion. It is used in a wide variety of security applications and Internet protocols.
To better understand some of the requirements and security implications for cryp-
tographic hash functions, it is useful to look at the range of applications in which it
is employed.

Message Authentication

Message authentication is a mechanism or service used to verify the integrity of
a message. Message authentication assures that data received are exactly as sent
(i.e., there is no modification, insertion, deletion, or replay). In many cases, there is
a requirement that the authentication mechanism assures that purported identity of
the sender is valid. When a hash function is used to provide message authentication,
the hash function value is often referred to as a message digest.1

The essence of the use of a hash function for message integrity is as follows.
The sender computes a hash value as a function of the bits in the message and trans-
mits both the hash value and the message. The receiver performs the same hash cal-
culation on the message bits and compares this value with the incoming hash value.

Figure 11.1 Cryptographic Hash Function; h = H(M)

Message or data block M (variable length) P, L

P, L = padding plus length field

L bits

Hash value h
(fixed length)

H

1The topic of this section is invariably referred to as message authentication. However, the concepts and
techniques apply equally to data at rest. For example, authentication techniques can be applied to a file
in storage to assure that the file is not tampered with.

342 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

If there is a mismatch, the receiver knows that the message (or possibly the hash
value) has been altered (Figure 11.2a).

The hash value must be transmitted in a secure fashion. That is, the hash value
must be protected so that if an adversary alters or replaces the message, it is not
feasible for adversary to also alter the hash value to fool the receiver. This type
of attack is shown in Figure 11.2b. In this example, Alice transmits a data block
and attaches a hash value. Darth intercepts the message, alters or replaces the data
block, and calculates and attaches a new hash value. Bob receives the altered data
with the new hash value and does not detect the change. To prevent this attack, the
hash value generated by Alice must be protected.

Figure 11.2 Attack Against Hash Function

(b) Man-in-the-middle attack

Alice

Darth

Bob

BobAlice

COMPARE

data

data

data

H

data

data

data

H

H

(a) Use of hash function to check data integrity

COMPARE

data

data

data

H
H

11.1 / APPLICATIONS OF CRYPTOGRAPHIC HASH FUNCTIONS 343

Figure 11.3 illustrates a variety of ways in which a hash code can be used to
provide message authentication, as follows.

a. The message plus concatenated hash code is encrypted using symmetric
encryption. Because only A and B share the secret key, the message must have
come from A and has not been altered. The hash code provides the structure or
redundancy required to achieve authentication. Because encryption is applied
to the entire message plus hash code, confidentiality is also provided.

b. Only the hash code is encrypted, using symmetric encryption. This reduces the
processing burden for those applications that do not require confidentiality.

Figure 11.3 Simplified Examples of the Use of a Hash Function for Message Authentication

E

K

M

H

| | D

K

M

H(M )

H

Compare

(a)

M

H

| |

K

(b)

M

D

H

CompareK

E

E(K, [M || H(M )])

E(K, H(M ))

Destination BSource A

| |S
M

H

| |

S
(c)

| |

M

H(M || S)

H(M || S)

H

Compare

M

H

| |

S
(d)

| |

E

K

| |S H

Compare

MD

K
E(K, [M || H(M || S)])

Hiva-Network.Com

344 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

c. It is possible to use a hash function but no encryption for message authentica-
tion. The technique assumes that the two communicating parties share a common
secret value S. A computes the hash value over the concatenation of M and S and
appends the resulting hash value to M. Because B possesses S, it can recompute
the hash value to verify. Because the secret value itself is not sent, an opponent
cannot modify an intercepted message and cannot generate a false message.

d. Confidentiality can be added to the approach of method (c) by encrypting the
entire message plus the hash code.

When confidentiality is not required, method (b) has an advantage over
methods (a) and (d), which encrypts the entire message, in that less computa-
tion is required. Nevertheless, there has been growing interest in techniques that
avoid encryption (Figure 11.3c). Several reasons for this interest are pointed out
in [TSUD92].

■ Encryption software is relatively slow. Even though the amount of data to be
encrypted per message is small, there may be a steady stream of messages into
and out of a system.

■ Encryption hardware costs are not negligible. Low-cost chip implementations
of DES are available, but the cost adds up if all nodes in a network must have
this capability.

■ Encryption hardware is optimized toward large data sizes. For small blocks of
data, a high proportion of the time is spent in initialization/invocation overhead.

■ Encryption algorithms may be covered by patents, and there is a cost associ-
ated with licensing their use.

More commonly, message authentication is achieved using a message
authentication code (MAC), also known as a keyed hash function. Typically, MACs
are used between two parties that share a secret key to authenticate information
exchanged between those parties. A MAC function takes as input a secret key and
a data block and produces a hash value, referred to as the MAC, which is associ-
ated with the protected message. If the integrity of the message needs to be checked,
the MAC function can be applied to the message and the result compared with the
associated MAC value. An attacker who alters the message will be unable to alter the
associated MAC value without knowledge of the secret key. Note that the verifying
party also knows who the sending party is because no one else knows the secret key.

Note that the combination of hashing and encryption results in an overall
function that is, in fact, a MAC (Figure 11.3b). That is, E(K, H(M)) is a function of
a variable-length message M and a secret key K, and it produces a fixed-size output
that is secure against an opponent who does not know the secret key. In practice,
specific MAC algorithms are designed that are generally more efficient than an
encryption algorithm.

We discuss MACs in Chapter 12.

Digital Signatures

Another important application, which is similar to the message authentication
application, is the digital signature. The operation of the digital signature is similar
to that of the MAC. In the case of the digital signature, the hash value of a message

11.1 / APPLICATIONS OF CRYPTOGRAPHIC HASH FUNCTIONS 345

is encrypted with a user’s private key. Anyone who knows the user’s public key can
verify the integrity of the message that is associated with the digital signature. In
this case, an attacker who wishes to alter the message would need to know the user’s
private key. As we shall see in Chapter 14, the implications of digital signatures go
beyond just message authentication.

Figure 11.4 illustrates, in a simplified fashion, how a hash code is used to
provide a digital signature.

a. The hash code is encrypted, using public-key encryption with the sender’s
private key. As with Figure 11.3b, this provides authentication. It also provides
a digital signature, because only the sender could have produced the encrypted
hash code. In fact, this is the essence of the digital signature technique.

b. If confidentiality as well as a digital signature is desired, then the message
plus the private-key-encrypted hash code can be encrypted using a symmetric
secret key. This is a common technique.

Other Applications

Hash functions are commonly used to create a one-way password file. Chapter 21
explains a scheme in which a hash of a password is stored by an operating system
rather than the password itself. Thus, the actual password is not retrievable by a
hacker who gains access to the password file. In simple terms, when a user enters a
password, the hash of that password is compared to the stored hash value for veri-
fication. This approach to password protection is used by most operating systems.

Hash functions can be used for intrusion detection and virus detection. Store
H(F) for each file on a system and secure the hash values (e.g., on a CD-R that is

Figure 11.4 Simplified Examples of Digital Signatures

M

H

| | E

E

K

D

K

M

D

H

Compare

(b) E(PRa, H(M ))

E(K, [M || E(PRa, H(M ))])

Destination BSource A

PRa

PRa

PUa

PUa

M

H

| |

(a)

M

E D

H

Compare

E(PRa, H(M ))

346 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

kept secure). One can later determine if a file has been modified by recomputing
H(F). An intruder would need to change F without changing H(F).

A cryptographic hash function can be used to construct a pseudorandom
function (PRF) or a pseudorandom number generator (PRNG). A common
application for a hash-based PRF is for the generation of symmetric keys. We discuss
this application in Chapter 12.

11.2 TWO SIMPLE HASH FUNCTIONS

To get some feel for the security considerations involved in cryptographic hash
functions, we present two simple, insecure hash functions in this section. All hash
functions operate using the following general principles. The input (message, file,
etc.) is viewed as a sequence of n -bit blocks. The input is processed one block at a
time in an iterative fashion to produce an n-bit hash function.

One of the simplest hash functions is the bit-by-bit exclusive-OR (XOR) of
every block. This can be expressed as

Ci = bi1 ⊕ bi2 ⊕ g ⊕ bim

where

Ci = ith bit of the hash code, 1 … i … n
m = number of [email protected] blocks in the input
bij = ith bit in jth block
⊕ = XOR operation

This operation produces a simple parity bit for each bit position and is known
as a longitudinal redundancy check. It is reasonably effective for random data as a
data integrity check. Each n-bit hash value is equally likely. Thus, the probability
that a data error will result in an unchanged hash value is 2-n. With more predict-
ably formatted data, the function is less effective. For example, in most normal text
files, the high-order bit of each octet is always zero. So if a 128-bit hash value is
used, instead of an effectiveness of 2-128, the hash function on this type of data has
an effectiveness of 2-112.

A simple way to improve matters is to perform a one-bit circular shift, or
rotation, on the hash value after each block is processed. The procedure can be
summarized as follows.

1. Initially set the n-bit hash value to zero.

2. Process each successive n-bit block of data as follows:

a. Rotate the current hash value to the left by one bit.
b. XOR the block into the hash value.

This has the effect of “randomizing” the input more completely and overcoming
any regularities that appear in the input. Figure 11.5 illustrates these two types of
hash functions for 16-bit hash values.

11.2 / TWO SIMPLE HASH FUNCTIONS 347

Although the second procedure provides a good measure of data integrity, it is
virtually useless for data security when an encrypted hash code is used with a plain-
text message, as in Figures 11.3b and 11.4a. Given a message, it is an easy matter
to produce a new message that yields that hash code: Simply prepare the desired
alternate message and then append an n-bit block that forces the new message plus
block to yield the desired hash code.

Although a simple XOR or rotated XOR (RXOR) is insufficient if only the
hash code is encrypted, you may still feel that such a simple function could be
useful when the message together with the hash code is encrypted (Figure 11.3a).
But you must be careful. A technique originally proposed by the National
Bureau of Standards used the simple XOR applied to 64-bit blocks of the mes-
sage and then an encryption of the entire message that used the cipher block
chaining (CBC) mode. We can define the scheme as follows: Given a message M
consisting of a sequence of 64-bit blocks X1, X2, c , XN, define the hash code

Figure 11.5 Two Simple Hash Functions

XOR of every 16-bit blockXOR with 1-bit r otation to the right

16 bits

348 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

h = H(M) as the block-by-block XOR of all blocks and append the hash code as
the final block:

h = XN + 1 = X1 ⊕ X2 ⊕ c ⊕ XN

Next, encrypt the entire message plus hash code using CBC mode to produce the
encrypted message Y1, Y2, c , YN + 1. [JUEN85] points out several ways in which
the ciphertext of this message can be manipulated in such a way that it is not detect-
able by the hash code. For example, by the definition of CBC (Figure 6.4), we have

X1 = IV⊕D(K,Y1)
Xi = Yi- 1⊕D(K, Yi)

XN + 1 = YN⊕D(K, YN + 1)

But XN + 1 is the hash code:

XN + 1 = X1⊕ X2⊕ c⊕ XN
= [IV⊕D(K, Y1)]⊕ [Y1⊕D(K, Y2)]⊕ c⊕ [YN - 1⊕D(K, YN)]

Because the terms in the preceding equation can be XORed in any order, it follows
that the hash code would not change if the ciphertext blocks were permuted.

11.3 REQUIREMENTS AND SECURITY

Before proceeding, we need to define two terms. For a hash value h = H(x), we
say that x is the preimage of h. That is, x is a data block whose hash value, using the
function H, is h. Because H is a many-to-one mapping, for any given hash value h,
there will in general be multiple preimages. A collision occurs if we have x ≠ y and
H(x) = H(y). Because we are using hash functions for data integrity, collisions are
clearly undesirable.

Let us consider how many preimages are there for a given hash value, which is
a measure of the number of potential collisions for a given hash value. Suppose the
length of the hash code is n bits, and the function H takes as input messages or data
blocks of length b bits with b 7 n. Then, the total number of possible messages is
2b and the total number of possible hash values is 2n. On average, each hash value
corresponds to 2b - n preimages. If H tends to uniformly distribute hash values then,
in fact, each hash value will have close to 2b - n preimages. If we now allow inputs
of arbitrary length, not just a fixed length of some number of bits, then the number
of preimages per hash value is arbitrarily large. However, the security risks in the
use of a hash function are not as severe as they might appear from this analysis.
To understand better the security implications of cryptographic hash functions, we
need precisely define their security requirements.

Security Requirements for Cryptographic Hash Functions

Table 11.1 lists the generally accepted requirements for a cryptographic hash func-
tion. The first three properties are requirements for the practical application of a
hash function.

11.3 / REQUIREMENTS AND SECURITY 349

The fourth property, preimage resistant, is the one-way property: it is easy
to generate a code given a message, but virtually impossible to generate a message
given a code. This property is important if the authentication technique involves the
use of a secret value (Figure 11.3c). The secret value itself is not sent. However, if
the hash function is not one way, an attacker can easily discover the secret value:
If the attacker can observe or intercept a transmission, the attacker obtains the
message M, and the hash code h = H(S }M). The attacker then inverts the hash
function to obtain S }M = H-1(MDM). Because the attacker now has both M and
SAB }M, it is a trivial matter to recover SAB.

The fifth property, second preimage resistant, guarantees that it is infeasible to
find an alternative message with the same hash value as a given message. This pre-
vents forgery when an encrypted hash code is used (Figures 11.3b and 11.4a). If this
property were not true, an attacker would be capable of the following sequence:
First, observe or intercept a message plus its encrypted hash code; second, generate
an unencrypted hash code from the message; third, generate an alternate message
with the same hash code.

A hash function that satisfies the first five properties in Table 11.1 is referred
to as a weak hash function. If the sixth property, collision resistant, is also satis-
fied, then it is referred to as a strong hash function. A strong hash function protects
against an attack in which one party generates a message for another party to sign.
For example, suppose Bob writes an IOU message, sends it to Alice, and she signs
it. Bob finds two messages with the same hash, one of which requires Alice to pay a
small amount and one that requires a large payment. Alice signs the first message,
and Bob is then able to claim that the second message is authentic.

Figure 11.6 shows the relationships among the three resistant properties.
A function that is collision resistant is also second preimage resistant, but the
reverse is not necessarily true. A function can be collision resistant but not preim-
age resistant and vice versa. A function can be preimage resistant but not second
preimage resistant and vice versa. See [MENE97] for a discussion.

Requirement Description

Variable input size H can be applied to a block of data of any size.

Fixed output size H produces a fixed-length output.

Efficiency H(x) is relatively easy to compute for any
given x, making both hardware and software
implementations practical.

Preimage resistant (one-way property) For any given hash value h, it is computationally
infeasible to find y such that H(y) = h.

Second preimage resistant (weak collision
resistant)

For any given block x, it is computationally
infeasible to find y ≠ x with H(y) = H(x).

Collision resistant (strong collision resistant) It is computationally infeasible to find any pair
(x, y) with x ≠ y, such that H(x) = H(y).

Pseudorandomness Output of H meets standard tests for
pseudorandomness.

Table 11.1 Requirements for a Cryptographic Hash Function H

350 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

Table 11.2 shows the resistant properties required for various hash function
applications.

The final requirement in Table 11.1, pseudorandomness, has not tradition-
ally been listed as a requirement of cryptographic hash functions but is more or
less implied. [JOHN05] points out that cryptographic hash functions are commonly
used for key derivation and pseudorandom number generation, and that in message
integrity applications, the three resistant properties depend on the output of the
hash function appearing to be random. Thus, it makes sense to verify that in fact a
given hash function produces pseudorandom output.

Brute-Force Attacks

As with encryption algorithms, there are two categories of attacks on hash func-
tions: brute-force attacks and cryptanalysis. A brute-force attack does not depend
on the specific algorithm but depends only on bit length. In the case of a hash func-
tion, a brute-force attack depends only on the bit length of the hash value. A crypt-
analysis, in contrast, is an attack based on weaknesses in a particular cryptographic
algorithm. We look first at brute-force attacks.

Figure 11.6 Relationship Among Hash Function Properties

Second
preimage resistant

Preimage
resistant

Collision
resistant

Preimage Resistant
Second Preimage

Resistant Collision Resistant

Hash + digital signature yes yes yes*

Intrusion detection and virus
detection

yes

Hash + symmetric encryption

One-way password file yes

MAC yes yes yes*

Table 11.2 Hash Function Resistance Properties Required for Various Data Integrity Applications

*Resistance required if attacker is able to mount a chosen message attack

11.3 / REQUIREMENTS AND SECURITY 351

PREIMAGE AND SECOND PREIMAGE ATTACKS For a preimage or second preimage
attack, an adversary wishes to find a value y such that H(y) is equal to a given hash
value h. The brute-force method is to pick values of y at random and try each value
until a collision occurs. For an m-bit hash value, the level of effort is proportional
to 2m. Specifically, the adversary would have to try, on average, 2m - 1 values of y to
find one that generates a given hash value h. This result is derived in Appendix U
[Equation (U.1)].

COLLISION RESISTANT ATTACKS For a collision resistant attack, an adversary wishes
to find two messages or data blocks, x and y, that yield the same hash function:
H(x) = H(y). This turns out to require considerably less effort than a preimage or
second preimage attack. The effort required is explained by a mathematical result
referred to as the birthday paradox. In essence, if we choose random variables from
a uniform distribution in the range 0 through N - 1, then the probability that a
repeated element is encountered exceeds 0.5 after 2N choices have been made.
Thus, for an m-bit hash value, if we pick data blocks at random, we can expect to
find two data blocks with the same hash value within 22m = 2m/2 attempts. The
mathematical derivation of this result is found in Appendix U.

Yuval proposed the following strategy to exploit the birthday paradox in a
collision resistant attack [YUVA79].

1. The source, A, is prepared to sign a legitimate message x by appending the
appropriate m-bit hash code and encrypting that hash code with A’s private
key (Figure 11.4a).

2. The opponent generates 2m/2 variations x′ of x, all of which convey essentially
the same meaning, and stores the messages and their hash values.

3. The opponent prepares a fraudulent message y for which A’s signature is
desired.

4. The opponent generates minor variations y′ of y, all of which convey essen-
tially the same meaning. For each y′, the opponent computes H(y′), checks
for matches with any of the H(x′) values, and continues until a match is found.
That is, the process continues until a y′ is generated with a hash value equal to
the hash value of one of the x′ values.

5. The opponent offers the valid variation to A for signature. This signature can
then be attached to the fraudulent variation for transmission to the intended
recipient. Because the two variations have the same hash code, they will pro-
duce the same signature; the opponent is assured of success even though the
encryption key is not known.

Thus, if a 64-bit hash code is used, the level of effort required is only on the
order of 232 [see Appendix U, Equation (U.7)].

The generation of many variations that convey the same meaning is not diffi-
cult. For example, the opponent could insert a number of “space-space- backspace”
character pairs between words throughout the document. Variations could then
be generated by substituting “space-backspace-space” in selected instances.

352 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

Alternatively, the opponent could simply reword the message but retain the
meaning. Figure 11.7 provides an example.

To summarize, for a hash code of length m, the level of effort required, as we
have seen, is proportional to the following.

Preimage resistant 2m

Second preimage resistant 2m

Collision resistant 2m/2

Figure 11.7 A Letter in 238 Variations

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11.3 / REQUIREMENTS AND SECURITY 353

If collision resistance is required (and this is desirable for a general-purpose
secure hash code), then the value 2m/2 determines the strength of the hash code
against brute-force attacks. Van Oorschot and Wiener [VANO94] presented
a design for a $10 million collision search machine for MD5, which has a 128-bit hash
length, that could find a collision in 24 days. Thus, a 128-bit code may be viewed as
inadequate. The next step up, if a hash code is treated as a sequence of 32 bits,
is a 160-bit hash length. With a hash length of 160 bits, the same search machine
would require over four thousand years to find a collision. With today’s technology,
the time would be much shorter, so that 160 bits now appears suspect.

Cryptanalysis

As with encryption algorithms, cryptanalytic attacks on hash functions seek to
exploit some property of the algorithm to perform some attack other than an
exhaustive search. The way to measure the resistance of a hash algorithm to crypt-
analysis is to compare its strength to the effort required for a brute-force attack.
That is, an ideal hash algorithm will require a cryptanalytic effort greater than or
equal to the brute-force effort.

In recent years, there has been considerable effort, and some successes,
in developing cryptanalytic attacks on hash functions. To understand these, we
need to look at the overall structure of a typical secure hash function, indicated
in Figure 11.8. This structure, referred to as an iterated hash function, was pro-
posed by Merkle [MERK79, MERK89] and is the structure of most hash func-
tions in use today, including SHA, which is discussed later in this chapter. The
hash function takes an input message and partitions it into L fixed-sized blocks
of b bits each. If necessary, the final block is padded to b bits. The final block
also includes the value of the total length of the input to the hash function. The
inclusion of the length makes the job of the opponent more difficult. Either the
opponent must find two messages of equal length that hash to the same value or
two messages of differing lengths that, together with their length values, hash to
the same value.

Figure 11.8 General Structure of Secure Hash Code

f fn n
n

IV =
CV0 CV1

b

n

CVL–1

CVLn

b

Y0 Y1 YL–1

IV = Initial value
CVi = Chaining variable
Yi = ith input block
f = Compression algorithm

L = Number of input blocks
n = Length of hash code
b = Length of input block

b

f

354 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

The hash algorithm involves repeated use of a compression function, f, that
takes two inputs (an n-bit input from the previous step, called the chaining variable,
and a b-bit block) and produces an n-bit output. At the start of hashing, the chaining
variable has an initial value that is specified as part of the algorithm. The final value
of the chaining variable is the hash value. Often, b 7 n; hence the term compression.
The hash function can be summarized as

CV0 = IV = initial [email protected] value
CVi = f(CVi- 1, Yi- 1) 1 … i … L

H(M) = CVL

where the input to the hash function is a message M consisting of the blocks
Y0, Y1, c , YL - 1.

The motivation for this iterative structure stems from the observation by
Merkle [MERK89] and Damgard [DAMG89] that if the length field is included in
the input, and if the compression function is collision resistant, then so is the resul-
tant iterated hash function.2 Therefore, the structure can be used to produce
a secure hash function to operate on a message of any length. The problem of
designing a secure hash function reduces to that of designing a collision-resistant
compression function that operates on inputs of some fixed size.

Cryptanalysis of hash functions focuses on the internal structure of f and is
based on attempts to find efficient techniques for producing collisions for a single
execution of f. Once that is done, the attack must take into account the fixed value
of IV. The attack on f depends on exploiting its internal structure. Typically, as with
symmetric block ciphers, f consists of a series of rounds of processing, so that the
attack involves analysis of the pattern of bit changes from round to round.

Keep in mind that for any hash function there must exist collisions, because
we are mapping a message of length at least equal to twice the block size b (because
we must append a length field) into a hash code of length n, where b Ú n. What is
required is that it is computationally infeasible to find collisions.

The attacks that have been mounted on hash functions are rather complex and
beyond our scope here. For the interested reader, [DOBB96] and [BELL97] are
recommended.

11.4 HASH FUNCTIONS BASED ON CIPHER BLOCK CHAINING

A number of proposals have been made for hash functions based on using a cipher
block chaining technique, but without using the secret key. One of the first such
proposals was that of Rabin [RABI78]. Divide a message M into fixed-size blocks
M1, M2, c , MN and use a symmetric encryption system such as DES to compute
the hash code G as

H0 = initial value
Hi = E(Mi, Hi- 1)
G = HN

2The converse is not necessarily true.

11.5 / SECURE HASH ALGORITHM (SHA) 355

This is similar to the CBC technique, but in this case, there is no secret key. As with
any hash code, this scheme is subject to the birthday attack, and if the encryp-
tion algorithm is DES and only a 64-bit hash code is produced, then the system
is vulnerable.

Furthermore, another version of the birthday attack can be used even if the
opponent has access to only one message and its valid signature and cannot obtain
multiple signings. Here is the scenario: We assume that the opponent intercepts
a message with a signature in the form of an encrypted hash code and that the
unencrypted hash code is m bits long.

1. Use the algorithm defined at the beginning of this subsection to calculate the
unencrypted hash code G.

2. Construct any desired message in the form Q1, Q2, c , QN - 2.
3. Compute Hi = E(Qi, Hi- 1) for 1 … i … (N - 2).
4. Generate 2m/2 random blocks; for each block X, compute E(X, HN - 2).

Generate an additional 2m/2 random blocks; for each block Y, compute D(Y,
G), where D is the decryption function corresponding to E.

5. Based on the birthday paradox, with high probability there will be an X and Y
such that E(X, HN - 2) = D(Y, G).

6. Form the message Q1, Q2, c , QN - 2, X, Y. This message has the hash code G
and therefore can be used with the intercepted encrypted signature.

This form of attack is known as a meet-in-the-middle-attack. A number of
researchers have proposed refinements intended to strengthen the basic block
chaining approach. For example, Davies and Price [DAVI89] describe the variation:

Hi = E(Mi, Hi- 1)⊕Hi- 1

Another variation, proposed in [MEYE88], is

Hi = E(Hi- 1, Mi)⊕Mi

However, both of these schemes have been shown to be vulnerable to a variety
of attacks [MIYA90]. More generally, it can be shown that some form of birthday
attack will succeed against any hash scheme involving the use of cipher block chain-
ing without a secret key, provided that either the resulting hash code is small enough
(e.g., 64 bits or less) or that a larger hash code can be decomposed into independent
subcodes [JUEN87].

Thus, attention has been directed at finding other approaches to hashing.
Many of these have also been shown to have weaknesses [MITC92].

11.5 SECURE HASH ALGORITHM (SHA)

In recent years, the most widely used hash function has been the Secure Hash
Algorithm (SHA). Indeed, because virtually every other widely used hash function
had been found to have substantial cryptanalytic weaknesses, SHA was more or
less the last remaining standardized hash algorithm by 2005. SHA was developed

356 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

by the National Institute of Standards and Technology (NIST) and published as a
federal information processing standard (FIPS 180) in 1993. When weaknesses were
discovered in SHA, now known as SHA-0, a revised version was issued as FIPS
180-1 in 1995 and is referred to as SHA-1. The actual standards document is entitled
“Secure Hash Standard.” SHA is based on the hash function MD4, and its design
closely models MD4.

SHA-1 produces a hash value of 160 bits. In 2002, NIST produced a revised
version of the standard, FIPS 180-2, that defined three new versions of SHA, with
hash value lengths of 256, 384, and 512 bits, known as SHA-256, SHA-384, and
SHA-512, respectively. Collectively, these hash algorithms are known as SHA-2.
These new versions have the same underlying structure and use the same types of
modular arithmetic and logical binary operations as SHA-1. A revised document
was issued as FIP PUB 180-3 in 2008, which added a 224-bit version (Table 11.3).
In 2015, NIST issued FIPS 180-4, which added two additional algorithms:
SHA-512/224 and SHA-512/256. SHA-1 and SHA-2 are also specified in RFC
6234, which essentially duplicates the material in FIPS 180-3 but adds a C code
implementation.

In 2005, NIST announced the intention to phase out approval of SHA-1 and
move to a reliance on SHA-2 by 2010. Shortly thereafter, a research team described
an attack in which two separate messages could be found that deliver the same
SHA-1 hash using 269 operations, far fewer than the 280 operations previously
thought needed to find a collision with an SHA-1 hash [WANG05]. This result
should hasten the transition to SHA-2.

In this section, we provide a description of SHA-512. The other versions are
quite similar.

SHA-512 Logic

The algorithm takes as input a message with a maximum length of less than 2128 bits
and produces as output a 512-bit message digest. The input is processed in 1024-bit
blocks. Figure 11.9 depicts the overall processing of a message to produce a digest.

Algorithm Message Size Block Size Word Size
Message

Digest Size

SHA-1 6 264 512 32 160

SHA-224 6 264 512 32 224

SHA-256 6 264 512 32 256

SHA-384 6 2128 1024 64 384

SHA-512 6 2128 1024 64 512

SHA-512/224 6 2128 1024 64 224

SHA-512/256 6 2128 1024 64 256

Note: All sizes are measured in bits.

Table 11.3 Comparison of SHA Parameters

11.5 / SECURE HASH ALGORITHM (SHA) 357

This follows the general structure depicted in Figure 11.8. The processing consists
of the following steps.

Step 1 Append padding bits. The message is padded so that its length is congruent
to 896 modulo 1024 [length K 896(mod 1024)]. Padding is always added,
even if the message is already of the desired length. Thus, the number of
padding bits is in the range of 1 to 1024. The padding consists of a single 1 bit
followed by the necessary number of 0 bits.

Step 2 Append length. A block of 128 bits is appended to the message. This block
is treated as an unsigned 128-bit integer (most significant byte first) and
contains the length of the original message in bits (before the padding).

The outcome of the first two steps yields a message that is an integer
multiple of 1024 bits in length. In Figure 11.9, the expanded message is rep-
resented as the sequence of 1024-bit blocks M1, M2, c , MN, so that the
total length of the expanded message is N * 1024 bits.

Step 3 Initialize hash buffer. A 512-bit buffer is used to hold intermediate and final
results of the hash function. The buffer can be represented as eight 64-bit
registers (a, b, c, d, e, f, g, h). These registers are initialized to the following
64-bit integers (hexadecimal values):

a = 6A09E667F3BCC908 e = 510E527FADE682D1

b = BB67AE8584CAA73B f = 9B05688C2B3E6C1F

c = 3C6EF372FE94F82B g = 1F83D9ABFB41BD6B

d = A54FF53A5F1D36F1 h = 5BE0CD19137E2179

Figure 11.9 Message Digest Generation Using SHA-512

N 1024 bits

M1

H1

M2 MN

F

IV = H0

Message

hash code

1024 bits

512 bits 512 bits 512 bits

1024 bits 1024 bits

L bits

L

128 bits

1000000, . . . ,0

+

H2

F

+

HN

F

+

+ = word-by-word addition mod 264

358 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

These values are stored in big-endian format, which is the most significant
byte of a word in the low-address (leftmost) byte position. These words
were obtained by taking the first sixty-four bits of the fractional parts of the
square roots of the first eight prime numbers.

Step 4 Process message in 1024-bit (128-byte) blocks. The heart of the algorithm is
a module that consists of 80 rounds; this module is labeled F in Figure 11.9.
The logic is illustrated in Figure 11.10.

Each round takes as input the 512-bit buffer value, abcdefgh, and
updates the contents of the buffer. At input to the first round, the buffer
has the value of the intermediate hash value, Hi- 1. Each round t makes
use of a 64-bit value Wt, derived from the current 1024-bit block being pro-
cessed (Mi). These values are derived using a message schedule described
subsequently. Each round also makes use of an additive constant Kt, where
0 … t … 79 indicates one of the 80 rounds. These words represent the first
64 bits of the fractional parts of the cube roots of the first 80 prime numbers.
The constants provide a “randomized” set of 64-bit patterns, which should
eliminate any regularities in the input data. Table 11.4 shows these constants
in hexadecimal format (from left to right).

Figure 11.10 SHA-512 Processing of a Single 1024-Bit Block

64

Mi

Wt

Hi

Hi–1

W0

W79

Kt

K0

K79

a b c

Round 0

d e f g h

a b c

Round t

d e f g h

Message
schedule

a b c

Round 79

d e f g h

+ + + + + + + +

11.5 / SECURE HASH ALGORITHM (SHA) 359

The output of the eightieth round is added to the input to the first
round (Hi- 1) to produce Hi. The addition is done independently for each of
the eight words in the buffer with each of the corresponding words in Hi- 1,
using addition modulo 264.

Step 5 Output. After all N 1024-bit blocks have been processed, the output from
the Nth stage is the 512-bit message digest.

We can summarize the behavior of SHA-512 as follows:

H0 = IV
Hi = SUM64(Hi- 1, abcdefghi)

MD = HN

where

IV = initial value of the abcdefgh buffer, defined in step 3
abcdefghi = the output of the last round of processing of the ith message block
N = the number of blocks in the message (including padding and

length fields)

SUM64 = addition modulo 264 performed separately on each word of the
pair of inputs

MD = final message digest value

428a2f98d728ae22 7137449123ef65cd b5c0fbcfec4d3b2f e9b5dba58189dbbc

3956c25bf348b538 59f111f1b605d019 923f82a4af194f9b ab1c5ed5da6d8118

d807aa98a3030242 12835b0145706fbe 243185be4ee4b28c 550c7dc3d5ffb4e2

72be5d74f27b896f 80deb1fe3b1696b1 9bdc06a725c71235 c19bf174cf692694

e49b69c19ef14ad2 efbe4786384f25e3 0fc19dc68b8cd5b5 240ca1cc77ac9c65

2de92c6f592b0275 4a7484aa6ea6e483 5cb0a9dcbd41fbd4 76f988da831153b5

983e5152ee66dfab a831c66d2db43210 b00327c898fb213f bf597fc7beef0ee4

c6e00bf33da88fc2 d5a79147930aa725 06ca6351e003826f 142929670a0e6e70

27b70a8546d22ffc 2e1b21385c26c926 4d2c6dfc5ac42aed 53380d139d95b3df

650a73548baf63de 766a0abb3c77b2a8 81c2c92e47edaee6 92722c851482353b

a2bfe8a14cf10364 a81a664bbc423001 c24b8b70d0f89791 c76c51a30654be30

d192e819d6ef5218 d69906245565a910 f40e35855771202a 106aa07032bbd1b8

19a4c116b8d2d0c8 1e376c085141ab53 2748774cdf8eeb99 34b0bcb5e19b48a8

391c0cb3c5c95a63 4ed8aa4ae3418acb 5b9cca4f7763e373 682e6ff3d6b2b8a3

748f82ee5defb2fc 78a5636f43172f60 84c87814a1f0ab72 8cc702081a6439ec

90befffa23631e28 a4506cebde82bde9 bef9a3f7b2c67915 c67178f2e372532b

ca273eceea26619c d186b8c721c0c207 eada7dd6cde0eb1e f57d4f7fee6ed178

06f067aa72176fba 0a637dc5a2c898a6 113f9804bef90dae 1b710b35131c471b

28db77f523047d84 32caab7b40c72493 3c9ebe0a15c9bebc 431d67c49c100d4c

4cc5d4becb3e42b6 597f299cfc657e2a 5fcb6fab3ad6faec 6c44198c4a475817

Table 11.4 SHA-512 Constants

360 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

SHA-512 Round Function

Let us look in more detail at the logic in each of the 80 steps of the processing
of one 512-bit block (Figure 11.11). Each round is defined by the following set of
equations:

T1 = h + Ch(e, f, g) + (a
512
1 e) + Wt + Kt

T2 = (a
512
0 a) + Maj(a, b, c)

h = g
g = f
f = e
e = d + T1
d = c
c = b
b = a
a = T1 + T2

where

t = step number; 0 … t … 79
Ch(e, f, g) = (e AND f)⊕ (NOT e AND g)

the conditional function: If e then f else g

Maj(a, b, c) = (a AND b)⊕ (a AND c)⊕ (b AND c)
the function is true only of the majority (two or three) of the

arguments are true

(Σ5120 a) = ROTR28(a)⊕ ROTR34(a)⊕ ROTR39(a)
(Σ5121 e) = ROTR14(e)⊕ ROTR18(e)⊕ ROTR41(e)
ROTRn(x) = circular right shift (rotation) of the 64-bit argument x by n bits

Figure 11.11 Elementary SHA-512 Operation (single round)

a b c d e f g h

a b c d
512 bits

e f g h

Ch

Kt

Wt

Maj

+

+
+

+

+

+

+

11.5 / SECURE HASH ALGORITHM (SHA) 361

Wt = a 64-bit word derived from the current 1024-bit input block
Kt = a 64-bit additive constant
+ = addition modulo 264

Two observations can be made about the round function.

1. Six of the eight words of the output of the round function involve simply per-
mutation (b, c, d, f, g, h) by means of rotation. This is indicated by shading in
Figure 11.11.

2. Only two of the output words (a, e) are generated by substitution. Word e is a
function of input variables (d, e, f, g, h), as well as the round word Wt and the
constant Kt. Word a is a function of all of the input variables except d, as well
as the round word Wt and the constant Kt.

It remains to indicate how the 64-bit word values Wt are derived from the
1024-bit message. Figure 11.12 illustrates the mapping. The first 16 values of Wt are
taken directly from the 16 words of the current block. The remaining values are
defined as

Wt = s1512(Wt- 2) + Wt- 7 + s0512(Wt- 15) + Wt- 16

where

s0
512(x) = ROTR1(x)⊕ ROTR8(x)⊕ SHR7(x)

s1
512(x) = ROTR19(x)⊕ ROTR61(x)⊕ SHR6(x)

ROTRn(x) = circular right shift (rotation) of the 64-bit argument x by n bits
SHRn(x) = right shift of the 64-bit argument x by n bits with padding by zeros on

the left
+ = addition modulo 264

Thus, in the first 16 steps of processing, the value of Wt is equal to the cor-
responding word in the message block. For the remaining 64 steps, the value of
Wt consists of the circular left shift by one bit of the XOR of four of the preced-
ing values of Wt, with two of those values subjected to shift and rotate operations.

Figure 11.12 Creation of 80-word Input Sequence for SHA-512 Processing of Single Block

1024 bits

64 bits

Wt–16W0 W1 W9 W14 W63 W64 W72 W77Wt–15 Wt–7 Wt–2

W0 W1 W15 W16 Wt

Mi

W79

+

s0 s1 s0 s1 s0 s1

+ +

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362 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

This introduces a great deal of redundancy and interdependence into the message
blocks that are compressed, which complicates the task of finding a different
message block that maps to the same compression function output. Figure 11.13
summarizes the SHA-512 logic.

The SHA-512 algorithm has the property that every bit of the hash code is a
function of every bit of the input. The complex repetition of the basic function F
produces results that are well mixed; that is, it is unlikely that two messages cho-
sen at random, even if they exhibit similar regularities, will have the same hash
code. Unless there is some hidden weakness in SHA-512, which has not so far been
published, the difficulty of coming up with two messages having the same message
digest is on the order of 2256 operations, while the difficulty of finding a message
with a given digest is on the order of 2512 operations.

Example

We include here an example based on one in FIPS 180. We wish to hash a one-block
message consisting of three ASCII characters: “abc,” which is equivalent to the
following 24-bit binary string:

01100001 01100010 01100011

Recall from step 1 of the SHA algorithm, that the message is padded to a
length congruent to 896 modulo 1024. In this case of a single block, the padding
consists of 896 - 24 = 872 bits, consisting of a “1” bit followed by 871 “0” bits.
Then a 128-bit length value is appended to the message, which contains the length
of the original message in bits (before the padding). The original length is 24 bits,
or a hexadecimal value of 18. Putting this all together, the 1024-bit message block,
in hexadecimal, is

6162638000000000 0000000000000000 0000000000000000 0000000000000000
0000000000000000 0000000000000000 0000000000000000 0000000000000000
0000000000000000 0000000000000000 0000000000000000 0000000000000000
0000000000000000 0000000000000000 0000000000000000 0000000000000018

This block is assigned to the words W0, . . . , W15 of the message schedule,
which appears as follows.

W0 = 6162638000000000 W8 = 0000000000000000
W1 = 0000000000000000 W9 = 0000000000000000
W2 = 0000000000000000 W10 = 0000000000000000
W3 = 0000000000000000 W11 = 0000000000000000
W4 = 0000000000000000 W12 = 0000000000000000
W5 = 0000000000000000 W13 = 0000000000000000
W6 = 0000000000000000 W14 = 0000000000000000
W7 = 0000000000000000 W15 = 0000000000000018

11.5 / SECURE HASH ALGORITHM (SHA) 363

The padded message consists blocks M1, M2, c , MN. Each message
block Mi consists of 16 64-bit words Mi,0, Mi,1, c , Mi,15. All addition
is performed modulo 264.

H0,0 = 6A09E667F3BCC908 H0,4 = 510E527FADE682D1
H0,1 = BB67AE8584CAA73B H0,5 = 9B05688C2B3E6C1F
H0,2 = 3C6EF372FE94F82B H0,6 = 1F83D9ABFB41BD6B
H0,3 = A54FF53A5F1D36F1 H0,7 = 5BE0CD19137E2179

for i = 1 to N
1. Prepare the message schedule W

for t = 0 to 15
Wt = Mi,t

for t = 16 to 79
Wt = s1512(Wt- 2) + Wt- 7 + s0512(Wt- 15) + Wt- 16

2. Initialize the working variables

a = Hi- 1, 0 e = Hi- 1, 4
b = Hi- 1, 1 f = Hi- 1, 5
c = Hi- 1, 2 g = Hi- 1, 6
d = Hi- 1, 3 h = Hi- 1, 7

3. Perform the main hash computation
for t = 0 to 79

T1 = h + Ch(e, f, g) + ¢Σ5121 e≤ + Wt + Kt
T2 = ¢Σ5120 a≤ + Maj(a, b, c)
h = g
g = f
f = e
e = d + T1
d = c
c = b
b = a
a = T1 + T2

4. Compute the intermediate hash value

Hi, 0 = a + Hi- 1, 0 Hi, 4 = e + Hi- 1,4
Hi, 1 = b + Hi- 1, 1 Hi, 5 = f + Hi- 1, 5
Hi, 2 = c + Hi- 1, 2 Hi, 6 = g + Hi- 1, 6
Hi, 3 = d + Hi- 1, 3 Hi, 7 = h + Hi- 1, 7

return {HN, 0 }HN, 1 }HN, 2 }HN, 3 }HN, 4 }HN, 5 }HN, 6 }HN, 7}

Figure 11.13 SHA-512 Logic

364 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

As indicated in Figure 11.13, the eight 64-bit variables, a through h, are
initialized to values H0,0 through H0,7. The following table shows the initial values of
these variables and their values after each of the first two rounds.

a 6a09e667f3bcc908 f6afceb8bcfcddf5 1320f8c9fb872cc0

b bb67ae8584caa73b 6a09e667f3bcc908 f6afceb8bcfcddf5

c 3c6ef372fe94f82b bb67ae8584caa73b 6a09e667f3bcc908

d a54ff53a5f1d36f1 3c6ef372fe94f82b bb67ae8584caa73b

e 510e527fade682d1 58cb02347ab51f91 c3d4ebfd48650ffa

f 9b05688c2b3e6c1f 510e527fade682d1 58cb02347ab51f91

g 1f83d9abfb41bd6b 9b05688c2b3e6c1f 510e527fade682d1

h 5be0cd19137e2179 1f83d9abfb41bd6b 9b05688c2b3e6c1f

Note that in each of the rounds, six of the variables are copied directly from
variables from the preceding round.

The process continues through 80 rounds. The output of the final round is

73a54f399fa4b1b2 10d9c4c4295599f6 d67806db8b148677 654ef9abec389ca9
d08446aa79693ed7 9bb4d39778c07f9e 25c96a7768fb2aa3 ceb9fc3691ce8326

The hash value is then calculated as

H1,0 = 6a09e667f3bcc908 + 73a54f399fa4b1b2 = ddaf35a193617aba
H1,1 = bb67ae8584caa73b + 10d9c4c4295599f6 = cc417349ae204131
H1,2 = 3c6ef372fe94f82b + d67806db8b148677 = 12e6fa4e89a97ea2
H1,3 = a54ff53a5f1d36f1 + 654ef9abec389ca9 = 0a9eeee64b55d39a
H1,4 = 510e527fade682d1 + d08446aa79693ed7 = 2192992a274fc1a8
H1,5 = 9b05688c2b3e6c1f + 9bb4d39778c07f9e = 36ba3c23a3feebbd
H1,6 = 1f83d9abfb41bd6b + 25c96a7768fb2aa3 = 454d4423643ce80e
H1,7 = 5be0cd19137e2179 + ceb9fc3691ce8326 = 2a9ac94fa54ca49f

The resulting 512-bit message digest is

ddaf35a193617aba cc417349ae204131 12e6fa4e89a97ea2 0a9eeee64b55d39a
2192992a274fc1a8 36ba3c23a3feebbd 454d4423643ce80e 2a9ac94fa54ca49f

Suppose now that we change the input message by one bit, from “abc” to
“cbc.” Then, the 1024-bit message block is

6362638000000000 0000000000000000 0000000000000000 0000000000000000
0000000000000000 0000000000000000 0000000000000000 0000000000000000
0000000000000000 0000000000000000 0000000000000000 0000000000000000
0000000000000000 0000000000000000 0000000000000000 0000000000000018

And the resulting 512-bit message digest is

531668966ee79b70 0b8e593261101354 4273f7ef7b31f279 2a7ef68d53f93264
319c165ad96d9187 55e6a204c2607e27 6e05cdf993a64c85 ef9e1e125c0f925f

The number of bit positions that differ between the two hash values is 253,
almost exactly half the bit positions, indicating that SHA-512 has a good avalanche
effect.

11.6 / SHA-3 365

11.6 SHA-3

As of this writing, the Secure Hash Algorithm (SHA-1) has not yet been “broken.”
That is, no one has demonstrated a technique for producing collisions in a practical
amount of time. However, because SHA-1 is very similar, in structure and in the
basic mathematical operations used, to MD5 and SHA-0, both of which have been
broken, SHA-1 is considered insecure and has been phased out for SHA-2.

SHA-2, particularly the 512-bit version, would appear to provide unassailable
security. However, SHA-2 shares the same structure and mathematical operations
as its predecessors, and this is a cause for concern. Because it will take years to find
a suitable replacement for SHA-2, should it become vulnerable, NIST decided to
begin the process of developing a new hash standard.

Accordingly, NIST announced in 2007 a competition to produce the next gen-
eration NIST hash function, to be called SHA-3. The winning design for SHA-3
was announced by NIST in October 2012 and published as FIP 102 in August 2015.
SHA-3 is a cryptographic hash function that is intended to complement SHA-2 as
the approved standard for a wide range of applications.

Appendix V looks at the evaluation criteria used by NIST to select from
among the candidates for AES, plus the rationale for picking Keccak, which was
the winning candidate. This material is useful in understanding not just the SHA-3
design but also the criteria by which to judge any cryptographic hash algorithm.

The Sponge Construction

The underlying structure of SHA-3 is a scheme referred to by its designers as a
sponge construction [BERT07, BERT11]. The sponge construction has the same
general structure as other iterated hash functions (Figure 11.8). The sponge func-
tion takes an input message and partitions it into fixed-size blocks. Each block is
processed in turn with the output of each iteration fed into the next iteration, finally
producing an output block.

The sponge function is defined by three parameters:

f = the internal function used to process each input block3

r = the size in bits of the input blocks, called the bitrate
pad = the padding algorithm

A sponge function allows both variable length input and output, making it a
flexible structure that can be used for a hash function (fixed-length output), a pseu-
dorandom number generator (fixed-length input), and other cryptographic func-
tions. Figure 11.14 illustrates this point. An input message of n bits is partitioned
into k fixed-size blocks of r bits each. The message is padded to achieve a length
that is an integer multiple of r bits. The resulting partition is the sequence of blocks
P0, P1, c , Pk - 1, with length k * r. For uniformity, padding is always added, so

3The Keccak documentation refers to f as a permutation. As we shall see, it involves both permutations
and substitutions. We refer to f as the iteration function, because it is the function that is executed once
for each iteration, that is, once for each block of the message that is processed.

366 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

that if n mod r = 0, a padding block of r bits is added. The actual padding algorithm
is a parameter of the function. The sponge specification [BERT11] proposes two
padding schemes:

■ Simple padding: Denoted by pad10*, appends a single bit 1 followed by the
minimum number of bits 0 such that the length of the result is a multiple of the
block length.

■ Multirate padding: Denoted by pad10*1, appends a single bit 1 followed by
the minimum number of bits 0 followed by a single bit 1 such that the length
of the result is a multiple of the block length. This is the simplest padding
scheme that allows secure use of the same f with different rates r. FIPS 202
uses multirate padding.

After processing all of the blocks, the sponge function generates a sequence
of output blocks Z0, Z1, c , Zj- 1. The number of output blocks generated is
determined by the number of output bits desired. If the desired output is / bits, then
j blocks are produced, such that (j - 1) * r 6 / … j * r.

Figure 11.15 shows the iterated structure of the sponge function. The sponge
construction operates on a state variable s of b = r + c bits, which is initialized
to all zeros and modified at each iteration. The value r is called the bitrate. This
value is the block size used to partition the input message. The term bitrate re-
flects the fact that r is the number of bits processed at each iteration: the larger the
value of r, the greater the rate at which message bits are processed by the sponge

Figure 11.14 Sponge Function Input and Output

k r bits

(a) Input

(b) Output

P0 P1

Z0 Z1

Zj–1

Pk–1

message pad

r bits r bits r bits

r bits r bits r bits

l bits

n bits

11.6 / SHA-3 367

construction. The value c is referred to as the capacity. A discussion of the secu-
rity implications of the capacity is beyond our scope. In essence, the capacity is a
measure of the achievable complexity of the sponge construction and therefore the
achievable level of security. A given implementation can trade claimed security for
speed by increasing the capacity c and decreasing the bitrate r accordingly, or vice
versa. The default values for Keccak are c = 1024 bits, r = 576 bits, and therefore
b = 1600 bits.

The sponge construction consists of two phases. The absorbing phase proceeds
as follows: For each iteration, the input block to be processed is padded with zeroes
to extend its length from r bits to b bits. Then, the bitwise XOR of the extended

Figure 11.15 Sponge Construction

(a) Absorbing phase

(b) Squeezing phase

f

r c

0c

0c

0c

0r 0c

P0

P1

P2

f

s

f

s

f

s

0cPk–1

b
r c

b

r c

Z0

r

Z1

368 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

message block and s is formed to create a b-bit input to the iteration function f. The
output of f is the value of s for the next iteration.

If the desired output length / satisfies / … b, then at the completion of the
absorbing phase, the first / bits of s are returned and the sponge construction termi-
nates. Otherwise, the sponge construction enters the squeezing phase. To begin, the
first / bits of s are retained as block Z0. Then, the value of s is updated with repeated
executions of f, and at each iteration, the first / bits of s are retained as block Zi
and concatenated with previously generated blocks. The process continues through
(j - 1) iterations until we have (j - 1) * r 6 / … j * r. At this point the first /
bits of the concatenated block Z are returned.

Note that the absorbing phase has the structure of a typical hash function.
A common case will be one in which the desired hash length is less than or equal
to the input block length; that is, / … r. In that case, the sponge construction termi-
nates after the absorbing phase. If a longer output than b bits is required, then the
squeezing phase is employed. Thus the sponge construction is quite flexible. For
example, a short message with a length r could be used as a seed and the sponge
construction would function as a pseudorandom number generator.

To summarize, the sponge construction is a simple iterated construction for
building a function F with variable-length input and arbitrary output length based
on a fixed-length transformation or permutation f operating on a fixed number b of
bits. The sponge construction is defined formally in [BERT11] as follows:

Algorithm The sponge construction SPONGE[f, pad, r]
Require: r < b

Interface: Z = sponge(M,/) with M ∈ Z2*, integer / > 0 and Z ∈ Z2/

P = M }pad[r](|M|)
s = 0b

for i = 0 to |P|r − 1 do
s = s⊕ (Pi }0b − r)
s = f(s)
end for
Z =:s;r
while |Z|r r < / do
s = f (s)
Z = Z } :s; r
end while
return :Z; ℓ

In the algorithm definition, the following notation is used: �M � is the length
in bits of a bit string M. A bit string M can be considered as a sequence of blocks
of some fixed length x, where the last block may be shorter. The number of
blocks of M is denoted by �M � x. The blocks of M are denoted by Mi and the index
ranges from 0 to �M � x - 1. The expression :M; / denotes the truncation of M to
its first / bits.

11.6 / SHA-3 369

Message Digest Size 224 256 384 512

Message Size no maximum no maximum no maximum no maximum

Block Size (bitrate r) 1152 1088 832 576

Word Size 64 64 64 64

Number of Rounds 24 24 24 24

Capacity c 448 512 768 1024

Collision Resistance 2112 2128 2192 2256

Second Preimage Resistance 2224 2256 2384 2512

Note: All sizes and security levels—are measured in bits.

Table 11.5 SHA-3 Parameters

SHA-3 makes use of the iteration function f, labeled Keccak-f, which is
described in the next section. The overall SHA-3 function is a sponge function
expressed as Keccak[r, c] to reflect that SHA-3 has two operational parameters, r,
the message block size, and c, the capacity, with the default of r + c = 1600 bits.
Table 11.5 shows the supported values of r and c. As Table 11.5 shows, the hash
function security associated with the sponge construction is a function of the
capacity c.

In terms of the sponge algorithm defined above, Keccak[r, c] is defined as

Keccak [r, c]∆ SPONGE [[email protected] [r + c], pad 10*1, r]

We now turn to a discussion of the iteration function Keccak-f.

The SHA-3 Iteration Function f

We now examine the iteration function Keccak-f used to process each successive
block of the input message. Recall that f takes as input a 1600-bit variable s consist-
ing of r bits, corresponding to the message block size followed by c bits, referred to
as the capacity. For internal processing within f, the input state variable s is orga-
nized as a 5 * 5 * 64 array a. The 64-bit units are referred to as lanes. For our
purposes, we generally use the notation a[x, y, z] to refer to an individual bit with
the state array. When we are more concerned with operations that affect entire
lanes, we designate the 5 * 5 matrix as L[x, y], where each entry in L is a 64-bit
lane. The use of indices within this matrix is shown in Figure 11.16.4 Thus, the col-
umns are labeled x = 0 through x = 4, the rows are labeled y = 0 through y = 4,
and the individual bits within a lane are labeled z = 0 through z = 63. The mapping
between the bits of s and those of a is

s[64(5y + x) + z] = a[x, y, z]

4Note that the first index (x) designates a column and the second index (y) designates a row. This is
in conflict with the convention used in most mathematics sources, where the first index designates a
row and the second index designates a column (e.g., Knuth, D. The Art of Computing Programming,
Volume 1, Fundamental Algorithms; and Korn, G., and Korn, T. Mathematical Handbook for Scientists
and Engineers).

370 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

We can visualize this with respect to the matrix in Figure 11.16. When treat-
ing the state as a matrix of lanes, the first lane in the lower left corner, L[0, 0], cor-
responds to the first 64 bits of s. The lane in the second column, lowest row, L[1,
0], corresponds to the next 64 bits of s. Thus, the array a is filled with the bits of s
starting with row y = 0 and proceeding row by row.

STRUCTURE OF f The function f is executed once for each input block of the message
to be hashed. The function takes as input the 1600-bit state variable and converts
it into a 5 * 5 matrix of 64-bit lanes. This matrix then passes through 24 rounds of
processing. Each round consists of five steps, and each step updates the state matrix
by permutation or substitution operations. As shown in Figure 11.17, the rounds are
identical with the exception of the final step in each round, which is modified by a
round constant that differs for each round.

The application of the five steps can be expressed as the composition5 of
functions:

R = i o x o p o r o u

Table 11.6 summarizes the operation of the five steps. The steps have a sim-
ple description leading to a specification that is compact and in which no trapdoor
can be hidden. The operations on lanes in the specification are limited to bitwise
Boolean operations (XOR, AND, NOT) and rotations. There is no need for table
lookups, arithmetic operations, or data-dependent rotations. Thus, SHA-3 is easily
and efficiently implemented in either hardware or software.

We examine each of the step functions in turn.

Figure 11.16 SHA-3 State Matrix

L[0, 4]

x = 0 x = 1 x = 2 x = 3 x = 4

L[0, 3]

L[0, 2]

L[0, 1]

L[0, 0]

a[x, y, 0] a[x, y, 1] a[x, y, 2]

y = 1

y = 0

y = 2

y = 3

y = 4 L[1, 4]

L[1, 3]

L[1, 2]

L[1, 1]

L[1, 0]

L[2, 4]

L[2, 3]

L[2, 2]

L[2, 1]

L[2, 0]

(a) State variable as 5 5 matrix A of 64-bit words

(b) Bit labeling of 64-bit words

L[3, 4]

L[3, 3]

L[3, 2]

L[4, 1]

L[3, 0]

L[4, 4]

L[4, 3]

L[4, 2]

L[4, 1]

L[4, 0]

a[x, y, 63]a[x, y, 62]a[x, y, z]

5If f and g are two functions, then the function F with the equation y = F(x) = g[f(x)] is called the
composition of f and g and is denoted as F = g o f.

Hiva-Network.Com

11.6 / SHA-3 371

Figure 11.17 SHA-3 Iteration Function f

theta (u) step

s

s

rho (r) step

pi (p) step

chi (x) step

R
ou

nd
0

iota (i) step RC[0]

rot(x, y)

theta (u) step

rho (r) step

pi (p) step

chi (x) step

R
ou

nd
2

3

iota (i) step RC[23]

rot(x, y)

Function Type Description

u Substitution New value of each bit in each word depends on its current
value and on one bit in each word of preceding column
and one bit of each word in succeeding column.

r Permutation The bits of each word are permuted using a circular bit
shift. W[0, 0] is not affected.

p Permutation Words are permuted in the 5 * 5 matrix. W[0, 0] is not
affected.

x Substitution New value of each bit in each word depends on its current
value and on one bit in next word in the same row and one
bit in the second next word in the same row.

i Substitution W[0, 0] is updated by XOR with a round constant.

Table 11.6 Step Functions in SHA-3

THETA STEP FUNCTION The Keccak reference defines the u function as follows. For
bit z in column x, row y,

u: a[x, y, z] d a[x, y, z]⊕ a
4

y==0
a[(x - 1), y=, z]⊕ a

4

y==0
a[(x + 1), y=, (z - 1)] (11.1)

372 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

where the summations are XOR operations. We can see more clearly what this
operation accomplishes with reference to Figure 11.18a. First, define the bitwise
XOR of the lanes in column x as

C[x] = L[x, 0]⊕ L[x, 1]⊕ L[x, 2]⊕ L[x, 3]⊕ L[x, 4]

Consider lane L[x, y] in column x, row y. The first summation in Equation 11.1
performs a bitwise XOR of the lanes in column (x - 1) mod 4 to form the 64-bit
lane C[x - 1]. The second summation performs a bitwise XOR of the lanes in
column (x + 1) mod 4, and then rotates the bits within the 64-bit lane so that the
bit in position z is mapped into position z + 1 mod 64. This forms the lane ROT
(C[x + 1], 1). These two lanes and L[x, y] are combined by bitwise XOR to form
the updated value of L[x, y]. This can be expressed as

L[x, y] d L[x, y]⊕ C[x - 1]⊕ ROT(C[x + 1], 1)

Figure 11.18.a illustrates the operation on L[3, 2]. The same operation is
performed on all of the other lanes in the matrix.

Figure 11.18 Theta and Chi Step Functions

(a) u step function

L[2, 3]L[2, 3] ROT(C[3], 1)C[1]

L[0, 4]

x = 0 x = 1 x = 2 x = 3 x = 4

L[0, 3]

L[0, 2]

L[0, 1]

L[0, 0]

y = 1

y = 0

y = 2

y = 3

y = 4 L[1, 4]

L[1, 3]

L[1, 2]

L[1, 1]

L[1, 0]

L[2, 4]

L[2, 3]

L[2, 2]

L[2, 1]

L[2, 0]

L[3, 4]

L[3, 3]

L[3, 2]

L[4, 1]

L[3, 0]

L[4, 4]

L[4, 3]

L[4, 2]

L[4, 1]

L[4, 0]

(b) x step function

L[2, 3]L[2, 3] L[3, 3] AND L[4, 3]

L[0, 4]

x = 0 x = 1 x = 2 x = 3 x = 4

L[0, 3]

L[0, 2]

L[0, 1]

L[0, 0]

y = 1

y = 0

y = 2

y = 3

y = 4 L[1, 4]

L[1, 3]

L[1, 2]

L[1, 1]

L[1, 0]

L[2, 4]

L[2, 3]

L[2, 2]

L[2, 1]

L[2, 0]

L[3, 4]

L[3, 3]

L[3, 2]

L[4, 1]

L[3, 0]

L[4, 4]

L[4, 3]

L[4, 2]

L[4, 1]

L[4, 0]

11.6 / SHA-3 373

Several observations are in order. Each bit in a lane is updated using the bit itself
and one bit in the same bit position from each lane in the preceding column and one
bit in the adjacent bit position from each lane in the succeeding column. Thus the up-
dated value of each bit depends on 11 bits. This provides good mixing. Also, the theta
step provides good diffusion, as that term was defined in Chapter 4. The designers of
Keccak state that the theta step provides a high level of diffusion on average and that
without theta, the round function would not provide diffusion of any significance.

RHO STEP FUNCTION The r function is defined as follows:

r: a[x, y, z] d a[x, y, z] if x = y = 0

otherwise,

r: a[x, y, z] d aJx, y, az - (t + 1)(t + 2)
2

b R (11.2)
with t satisfying 0 … t 6 24 and ¢0 1

2 3
≤t¢1

0
≤ = ¢x

y
≤ in GF(5)2 * 2

It is not immediately obvious what this step performs, so let us look at the
process in detail.

1. The lane in position (x, y) = (0, 0), that is L[0, 0], is unaffected. For all other
words, a circular bit shift within the lane is performed.

2. The variable t, with 0 … t 6 24, is used to determine both the amount of the
circular bit shift and which lane is assigned which shift value.

3. The 24 individual bit shifts that are performed have the respective values

(t + 1)(t + 2)
2

mod 64.

4. The shift determined by the value of t is performed on the lane in position
(x, y) in the 5 * 5 matrix of lanes. Specifically, for each value of t, the corre-

sponding matrix position is defined by ¢x
y
≤ = ¢0 1

2 3
≤t¢1

0
≤. For example, for

t = 3, we have

¢x
y
≤ = ¢0 1

2 3
≤3¢1

0
≤ mod 5

= ¢0 1
2 3

≤ ¢0 1
2 3

≤ ¢0 1
2 3

≤ ¢1
0
≤ mod 5

= ¢0 1
2 3

≤ ¢0 1
2 3

≤ ¢0
2
≤ mod 5

= ¢0 1
2 3

≤ ¢2
6
≤ mod 5 = ¢0 1

2 3
≤ ¢2

1
≤ mod 5

= ¢1
7
≤ mod 5 = ¢1

2
≤

374 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

Table 11.7 shows the calculations that are performed to determine the amount
of the bit shift and the location of each bit shift value. Note that all of the rotation
amounts are different.

The r function thus consists of a simple permutation (circular shift) within
each lane. The intent is to provide diffusion within each lane. Without this function,
diffusion between lanes would be very slow.

PI STEP FUNCTION The p function is defined as follows:

p: a[x, y] d a[x=, y=], with¢x
y
≤ = ¢0 1

2 3
≤ ¢x=

y=
≤ (11.3)

This can be rewritten as (x, y) * (y, (2x + 3y)). Thus, the lanes within the
5 * 5 matrix are moved so that the new x position equals the old y position and the

Table 11.7 Rotation Values Used in SHA-3

t g(t) g (t) mod 64 x, y

0 1 1 1, 0

1 3 3 0, 2

2 6 6 2, 1

3 10 10 1, 2

4 15 15 2, 3

5 21 21 3, 3

6 28 28 3, 0

7 36 36 0, 1

8 45 45 1, 3

9 55 55 3, 1

10 66 2 1, 4

11 78 14 4, 4

(b) Rotation values by word position in matrix

x = 0 x = 1 x = 2 x = 3 x = 4

y = 4 18 2 61 56 14

y = 3 41 45 15 21 8

y = 2 3 10 43 25 39

y = 1 36 44 6 55 20

y = 0 0 1 62 28 27

t g(t) g (t) mod 64 x, y

12 91 27 4, 0

13 105 41 0, 3

14 120 56 3, 4

15 136 8 4, 3

16 153 25 3, 2

17 171 43 2, 2

18 190 62 2, 0

19 210 18 0, 4

20 231 39 4, 2

21 253 61 2, 4

22 276 20 4, 1

23 300 44 1, 1

(a) Calculation of values and positions

Note: g(t) = (t + 1)(t + 2)/2

¢x
y
≤ = ¢0 1

2 3
≤t¢1

0
≤ mod 5

11.6 / SHA-3 375

Figure 11.19 Pi Step Function

Z[0, 4]

x = 0 x = 1 x = 2

(a) Lane position at start of step

(b) Lane position after permutation

x = 3 x = 4

Z[0, 3]

Z[0, 2]

Z[0, 1]

Z[0, 0]

y = 1

y = 0

y = 2

y = 3

y = 4 Z[1, 4]

Z[1, 3]

Z[1, 2]

Z[1, 1]

Z[1, 0]

Z[2, 4]

Z[2, 3]

Z[2, 2]

Z[2, 1]

Z[2, 0]

Z[3, 4]

Z[3, 3]

Z[3, 2]

Z[3, 1]

Z[3, 0]

Z[4, 4]

row
0row

3
row

1
row

4
row

2

row
2

row
4

row
1

row
3

Z[4, 3]

Z[4, 2]

Z[4, 1]

Z[4, 0]

Z[2, 0]

x = 0 x = 1 x = 2 x = 3 x = 4

Z[4, 0]

Z[1, 0]

Z[3, 0]

Z[0, 0]

y = 1

y = 0

y = 2

y = 3

y = 4 Z[3, 1]

Z[0, 1]

Z[2, 1]

Z[4, 1]

Z[1, 1]

Z[4, 2]

Z[1, 2]

Z[3, 2]

Z[0, 2]

Z[2, 2]

Z[0, 3]

Z[2, 3]

Z[4, 3]

Z[1, 3]

Z[3, 3]

Z[1, 4]

Z[3, 4]

Z[0, 4]

Z[2, 4]

Z[4, 4]

new y position is determined by (2x + 3y) mod 5. Figure 11.19 helps in visualizing
this permutation. Lanes that are along the same diagonal (increasing in y value,
going from left to right) prior to p are arranged on the same row in the matrix after
p is executed. Note that the position of L[0, 0] is unchanged.

Thus the p step is a permutation of lanes: The lanes move position within the
5 * 5 matrix. The r step is a permutation of bits: Bits within a lane are rotated.
Note that the p step matrix positions are calculated in the same way that, for the r
step, the one-dimensional sequence of rotation constants is mapped to the lanes of
the matrix.

CHI STEP FUNCTION The x function is defined as follows:

x: a[x] d a[x]⊕ ((a[x + 1]⊕ 1) AND a[x + 2]) (11.4)

This function operates to update each bit based on its current value and the
value of the corresponding bit position in the next two lanes in the same row. The

376 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

Round
Constant

(hexadecimal)
Number
of 1 bits

0 0000000000000001 1

1 0000000000008082 3

2 800000000000808A 5

3 8000000080008000 3

4 000000000000808B 5

5 0000000080000001 2

6 8000000080008081 5

7 8000000000008009 4

8 000000000000008A 3

9 0000000000000088 2

10 0000000080008009 4

11 000000008000000A 3

Table 11.8 Round Constants in SHA-3

Round
Constant

(hexadecimal)
Number
of 1 bits

12 000000008000808B 6

13 800000000000008B 5

14 8000000000008089 5

15 8000000000008003 4

16 8000000000008002 3

17 8000000000000080 2

18 000000000000800A 3

19 800000008000000A 4

20 8000000080008081 5

21 8000000000008080 3

22 0000000080000001 2

23 8000000080008008 4

operation is more clearly seen if we consider a single bit a[x, y, z] and write out the
Boolean expression:

a[x, y, z] d a[x, y, z]⊕ (NOT(a[x + 1, y, z])) AND (a[x + 2, y, z])

Figure 11.18b illustrates the operation of the x function on the bits of the
lane L[3, 2]. This is the only one of the step functions that is a nonlinear mapping.
Without it, the SHA-3 round function would be linear.

IOTA STEP FUNCTION The i function is defined as follows:

i: a d a⊕ RC[ir] (11.5)

This function combines an array element with a round constant that differs for
each round. It breaks up any symmetry induced by the other four step functions. In
fact, Equation 11.5 is somewhat misleading. The round constant is applied only to
the first lane of the internal state array. We express this is as follows:

L[0, 0] d L[0, 0]⊕ RC[ir] 0 … ir … 24

Table 11.8 lists the 24 64-bit round constants. Note that the Hamming weight,
or number of 1 bits, in the round constants ranges from 1 to 6. Most of the bit posi-
tions are zero and thus do not change the corresponding bits in L[0, 0]. If we take
the cumulative OR of all 24 round constants, we get

RC[0] OR RC[1] OR c OR RC[23] = 800000008000808B

Thus, only 7 bit positions are active and can affect the value of L[0, 0].
Of course, from round to round, the permutations and substitutions propagate the
effects of the i function to all of the lanes and all of the bit positions in the matrix.
It is easily seen that the disruption diffuses through u and x to all lanes of the state
after a single round.

11.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 377

11.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

absorbing phase
big endian
birthday attack
birthday paradox
bitrate
capacity
Chi step function collision

resistant
compression function
cryptographic hash function
hash code
hash function
hash value

Iota step function
Keccak
keyed hash function
lane
little endian
MD4
MD5
message authentication code

(MAC)
message digest
one-way hash function
Pi step function
preimage resistant

Rho step function
second preimage resistant
SHA-1
SHA-224
SHA-256
SHA-3
SHA-384
SHA-512
sponge construction
squeezing phase
strong collision resistance
Theta step function
weak collision resistance

Key Terms

Review Questions
11.1 What characteristics are needed in a secure hash function?
11.2 What is the difference between weak and strong collision resistance?
11.3 What is the role of a compression function in a hash function?
11.4 What is the difference between little-endian and big-endian format?
11.5 What basic arithmetical and logical functions are used in SHA?
11.6 Describe the set of criteria used by NIST to evaluate SHA-3 candidates.
11.7 Define the term sponge construction.
11.8 Briefly describe the internal structure of the iteration function f.
11.9 List and briefly describe the step functions that comprise the iteration function f.

Problems
11.1 The high-speed transport protocol XTP (Xpress Transfer Protocol) uses a 32-bit

checksum function defined as the concatenation of two 16-bit functions: XOR and
RXOR, defined in Section 11.4 as “two simple hash functions” and illustrated in
Figure 11.5.
a. Will this checksum detect all errors caused by an odd number of error bits?

Explain.
b. Will this checksum detect all errors caused by an even number of error bits? If not,

characterize the error patterns that will cause the checksum to fail.
c. Comment on the effectiveness of this function for use as a hash function for

authentication.
11.2 a. Consider the Davies and Price hash code scheme described in Section 11.4 and

assume that DES is used as the encryption algorithm:

Hi = Hi- 1⊕ E(Mi, Hi- 1)

378 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

Recall the complementarity property of DES (Problem 3.14): If Y = E(K, X),
then Y′ = E(K′, X′). Use this property to show how a message consisting of
blocks M1, M2, c , MN can be altered without altering its hash code.

b. Show that a similar attack will succeed against the scheme proposed in [MEYE88]:

Hi = Mi⊕ E(Hi- 1, Mi)

11.3 a. Consider the following hash function. Messages are in the form of a sequence of

numbers in Zn, M = (a1, a2, c at). The hash value h is calculated as ¢at
i=1

ai≤ for
some predefined value n. Does this hash function satisfy any of the requirements
for a hash function listed in Table 11.1? Explain your answer.

b. Repeat part (a) for the hash function h = ¢at
i=1

(ai)
2≤ mod n.

c. Calculate the hash function of part (b) for M = (189, 632, 900, 722, 349) and
n = 989.

11.4 It is possible to use a hash function to construct a block cipher with a structure similar
to DES. Because a hash function is one way and a block cipher must be reversible (to
decrypt), how is it possible?

11.5 Now consider the opposite problem: using an encryption algorithm to construct
a one-way hash function. Consider using RSA with a known key. Then process a
message consisting of a sequence of blocks as follows: Encrypt the first block, XOR
the result with the second block and encrypt again, etc. Show that this scheme is not
secure by solving the following problem. Given a two-block message B1, B2, and
its hash

RSAH(B1,B2) = RSA(RSA(B1)⊕ B2)

Given an arbitrary block C1, choose C2 so that RSAH(C1, C2) = RSAH(B1, B2).
Thus, the hash function does not satisfy weak collision resistance.

11.6 Suppose H(m) is a collision-resistant hash function that maps a message of arbitrary
bit length into an n-bit hash value. Is it true that, for all messages x, x′ with x ≠ x′,
we have H(x) ≠ H(x′) Explain your answer.

11.7 In Figure 11.12, it is assumed that an array of 80 64-bit words is available to store the
values of Wt, so that they can be precomputed at the beginning of the processing of
a block. Now assume that space is at a premium. As an alternative, consider the use
of a 16-word circular buffer that is initially loaded with W0 through W15. Design an
algorithm that, for each step t, computes the required input value Wt.

11.8 For SHA-512, show the equations for the values of W16, W18, W23, and W31.
11.9 State the value of the padding field in SHA-512 if the length of the message is

a. 2942 bits
b. 2943 bits
c. 2944 bits

11.10 State the value of the length field in SHA-512 if the length of the message is
a. 2942 bits
b. 2943 bits
c. 2944 bits

11.11 Suppose a1a2a3a4 are the 4 bytes in a 32-bit word. Each ai can be viewed as an integer
in the range 0 to 255, represented in binary. In a big-endian architecture, this word
represents the integer

a12
24 + a2216 + a328 + a4

11.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 379

In a little-endian architecture, this word represents the integer

a42
24 + a3216 + a228 + a1

a. Some hash functions, such as MD5, assume a little-endian architecture. It is impor-
tant that the message digest be independent of the underlying architecture. There-
fore, to perform the modulo 2 addition operation of MD5 or RIPEMD-160 on
a big-endian architecture, an adjustment must be made. Suppose X = x1 x2 x3 x4
and Y = y1 y2 y3 y4. Show how the MD5 addition operation (X + Y) would be
carried out on a big-endian machine.

b. SHA assumes a big-endian architecture. Show how the operation (X + Y) for
SHA would be carried out on a little-endian machine.

11.12 This problem introduces a hash function similar in spirit to SHA that operates on
letters instead of binary data. It is called the toy tetragraph hash (tth).6 Given a mes-
sage consisting of a sequence of letters, tth produces a hash value consisting of four
letters. First, tth divides the message into blocks of 16 letters, ignoring spaces, punc-
tuation, and capitalization. If the message length is not divisible by 16, it is padded
out with nulls. A four-number running total is maintained that starts out with the
value (0, 0, 0, 0); this is input to the compression function for processing the first
block. The compression function consists of two rounds.

Round 1 Get the next block of text and arrange it as a row-wise 4 * 4 block of text
and convert it to numbers (A = 0, B = 1, etc.). For example, for the block
ABCDEFGHIJKLMNOP, we have

A B C D

E F G H

I J K L

M N O P

0 1 2 3

4 5 6 7

8 9 10 11

12 13 14 15

Then, add each column mod 26 and add the result to the running total, mod 26. In this
example, the running total is (24, 2, 6, 10).

Round 2 Using the matrix from round 1, rotate the first row left by 1, second row left by 2,
third row left by 3, and reverse the order of the fourth row.
In our example:

B C D A

G H E F

L I J K

P O N M

1 2 3 0

6 7 4 5

11 8 9 10

15 14 13 12

Now, add each column mod 26 and add the result to the running total. The new run-
ning total is (5, 7, 9, 11). This running total is now the input into the first round of the
compression function for the next block of text. After the final block is processed,
convert the final running total to letters. For example, if the message is ABCDEF-
GHIJKLMNOP, then the hash is FHJL.

6I thank William K. Mason, of the magazine staff of The Cryptogram, for providing this example.

Hiva-Network.Com

380 CHAPTER 11 / CRYPTOGRAPHIC HASH FUNCTIONS

a. Draw figures comparable to Figures 11.9 and 11.10 to depict the overall tth logic
and the compression function logic.

b. Calculate the hash function for the 22-letter message “Practice makes us perfect.”
c. To demonstrate the weakness of tth, find a message of length 32-letter to produces

the same hash.
11.13 For each of the possible capacity values of SHA-3 (Table 11.5), which lanes in the

internal 55 state matrix start out as lanes of all zeros?
11.14 Consider the SHA-3 option with a block size of 1024 bits and assume that each of the

lanes in the first message block (P0) has at least one nonzero bit. To start, all of the
lanes in the internal state matrix that correspond to the capacity portion of the initial
state are all zeros. Show how long it will take before all of these lanes have at least
one nonzero bit. Note: Ignore the permutation. That is, keep track of the original zero
lanes even after they have changed position in the matrix.

11.15 Consider the state matrix as illustrated in Figure 11.16a. Now rearrange the rows and
columns of the matrix so that L[0, 0] is in the center. Specifically, arrange the columns
in the left-to-right order (x = 3, x = 4, x = 0, x = 1, x = 2) and arrange the rows in
the top-to-bottom order (y = 2, y = 1, y = 0, y = 4, y = 6). This should give you
some insight into the permutation algorithm used for the function and for permut-
ing the rotation constants in the function. Using this rearranged matrix, describe the
permutation algorithm.

11.16 The function only affects L[0, 0]. Section 11.6 states that the changes to L[0, 0] diffuse
through u and to all lanes of the state after a single round.
a. Show that this is so.
b. How long before all of the bit positions in the matrix are affected by the changes

to L[0, 0]?

381

Message Authentication
Codes

12.1 Message Authentication Requirements

12.2 Message Authentication Functions

Message Encryption
Message Authentication Code

12.3 Requirements for Message Authentication Codes

12.4 Security of MACs

Brute-Force Attacks
Cryptanalysis

12.5 MACs Based on Hash Functions: HMAC

HMAC Design Objectives
HMAC Algorithm
Security of HMAC

12.6 MACs Based on Block Ciphers: DAA and CMAC

Data Authentication Algorithm
Cipher-Based Message Authentication Code (CMAC)

12.7 Authenticated Encryption: CCM and GCM

Counter with Cipher Block Chaining-Message Authentication Code
Galois/Counter Mode

12.8 Key Wrapping

Background
The Key Wrapping Algorithm
Key Unwrapping

12.9 Pseudorandom Number Generation Using Hash Functions and MACs

PRNG Based on Hash Function
PRNG Based on MAC Function

12.10 Key Terms, Review Questions, and Problems

CHAPTER

382 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

One of the most fascinating and complex areas of cryptography is that of message
authentication and the related area of digital signatures. It would be impossible, in
anything less than book length, to exhaust all the cryptographic functions and proto-
cols that have been proposed or implemented for message authentication and digital
signatures. Instead, the purpose of this chapter and the next is to provide a broad
overview of the subject and to develop a systematic means of describing the various
approaches.

This chapter begins with an introduction to the requirements for authen-
tication and digital signature and the types of attacks to be countered. Then the
basic approaches are surveyed. The remainder of the chapter deals with the funda-
mental approach to message authentication known as the message authentication
code (MAC). Following an overview of this topic, the chapter looks at security
considerations for MACs. This is followed by a discussion of specific MACs in
two categories: those built from cryptographic hash functions and those built using
a block cipher mode of operation. Next, we look at a relatively recent approach
known as authenticated encryption. Finally, we look at the use of cryptographic
hash functions and MACs for pseudorandom number generation.

12.1 MESSAGE AUTHENTICATION REQUIREMENTS

In the context of communications across a network, the following attacks can be
identified.

1. Disclosure: Release of message contents to any person or process not possess-
ing the appropriate cryptographic key.

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ List and explain the possible attacks that are relevant to message
authentication.

◆ Define the term message authentication code.

◆ List and explain the requirements for a message authentication code.

◆ Present an overview of HMAC.

◆ Present an overview of CMAC.

◆ Explain the concept of authenticated encryption.

◆ Present an overview of CCM.

◆ Present an overview of GCM.

◆ Discuss the concept of key wrapping and explain its use.

◆ Understand how a hash function or a message authentication code can be
used for pseudorandom number generation.

12.2 / MESSAGE AUTHENTICATION FUNCTIONS 383

2. Traffic analysis: Discovery of the pattern of traffic between parties. In a
connection-oriented application, the frequency and duration of connec-
tions could be determined. In either a connection-oriented or connectionless
environment, the number and length of messages between parties could be
determined.

3. Masquerade: Insertion of messages into the network from a fraudulent source.
This includes the creation of messages by an opponent that are purported to
come from an authorized entity. Also included are fraudulent acknowledg-
ments of message receipt or nonreceipt by someone other than the message
recipient.

4. Content modification: Changes to the contents of a message, including inser-
tion, deletion, transposition, and modification.

5. Sequence modification: Any modification to a sequence of messages between
parties, including insertion, deletion, and reordering.

6. Timing modification: Delay or replay of messages. In a connection-oriented
application, an entire session or sequence of messages could be a replay of
some previous valid session, or individual messages in the sequence could be
delayed or replayed. In a connectionless application, an individual message
(e.g., datagram) could be delayed or replayed.

7. Source repudiation: Denial of transmission of message by source.

8. Destination repudiation: Denial of receipt of message by destination.

Measures to deal with the first two attacks are in the realm of message
confidentiality and are dealt with in Part One. Measures to deal with items
(3) through (6) in the foregoing list are generally regarded as message authentica-
tion. Mechanisms for dealing specifically with item (7) come under the heading of
digital signatures. Generally, a digital signature technique will also counter some
or all of the attacks listed under items (3) through (6). Dealing with item (8) may
require a combination of the use of digital signatures and a protocol designed to
counter this attack.

In summary, message authentication is a procedure to verify that received
messages come from the alleged source and have not been altered. Message au-
thentication may also verify sequencing and timeliness. A digital signature is an
authentication technique that also includes measures to counter repudiation by the
source.

12.2 MESSAGE AUTHENTICATION FUNCTIONS

Any message authentication or digital signature mechanism has two levels of func-
tionality. At the lower level, there must be some sort of function that produces an
authenticator: a value to be used to authenticate a message. This lower-level func-
tion is then used as a primitive in a higher-level authentication protocol that enables
a receiver to verify the authenticity of a message.

This section is concerned with the types of functions that may be used to pro-
duce an authenticator. These may be grouped into three classes.

384 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

■ Hash function: A function that maps a message of any length into a fixed-length
hash value, which serves as the authenticator

■ Message encryption: The ciphertext of the entire message serves as its
authenticator

■ Message authentication code (MAC): A function of the message and a secret
key that produces a fixed-length value that serves as the authenticator

Hash functions, and how they may serve for message authentication, are dis-
cussed in Chapter 11. The remainder of this section briefly examines the remaining
two topics. The remainder of the chapter elaborates on the topic of MACs.

Message Encryption

Message encryption by itself can provide a measure of authentication. The analysis
differs for symmetric and public-key encryption schemes.

SYMMETRIC ENCRYPTION Consider the straightforward use of symmetric encryption
(Figure 12.1a). A message M transmitted from source A to destination B is encrypted
using a secret key K shared by A and B. If no other party knows the key, then confi-
dentiality is provided: No other party can recover the plaintext of the message.

Figure 12.1 Basic Uses of Message Encryption

Destination BSource A

M

K K

E

(a) Symmetric encryption: confidentiality and authentication

D M

PUb
(b) Public-key encryption: confidentiality

E(K, M)

M E D M

E(PUb, M)

E(PRa, M) E(PRa, M)E(PUb, E(PRa, M))

M E D M

(c) Public-key encryption: authentication and signature

(d) Public-key encryption: confidentiality, authentication, and signature

E D

PRb

PRa

M E D M

E(PRa, M)

PRa

PUa

PUaPUb PRb

12.2 / MESSAGE AUTHENTICATION FUNCTIONS 385

In addition, B is assured that the message was generated by A. Why? The
message must have come from A, because A is the only other party that possesses
K and therefore the only other party with the information necessary to construct
ciphertext that can be decrypted with K. Furthermore, if M is recovered, B knows
that none of the bits of M have been altered, because an opponent that does not
know K would not know how to alter bits in the ciphertext to produce the desired
changes in the plaintext.

So we may say that symmetric encryption provides authentication as well as
confidentiality. However, this flat statement needs to be qualified. Consider exactly
what is happening at B. Given a decryption function D and a secret key K, the
destination will accept any input X and produce output Y = D(K, X). If X is the
ciphertext of a legitimate message M produced by the corresponding encryption
function, then Y is some plaintext message M. Otherwise, Y will likely be a mean-
ingless sequence of bits. There may need to be some automated means of determin-
ing at B whether Y is legitimate plaintext and therefore must have come from A.

The implications of the line of reasoning in the preceding paragraph are pro-
found from the point of view of authentication. Suppose the message M can be any
arbitrary bit pattern. In that case, there is no way to determine automatically, at the
destination, whether an incoming message is the ciphertext of a legitimate message.
This conclusion is incontrovertible: If M can be any bit pattern, then regardless of
the value of X, the value Y = D(K, X) is some bit pattern and therefore must be
accepted as authentic plaintext.

Thus, in general, we require that only a small subset of all possible bit patterns
be considered legitimate plaintext. In that case, any spurious ciphertext is unlikely
to produce legitimate plaintext. For example, suppose that only one bit pattern in
106 is legitimate plaintext. Then the probability that any randomly chosen bit pat-
tern, treated as ciphertext, will produce a legitimate plaintext message is only 10-6.

For a number of applications and encryption schemes, the desired conditions
prevail as a matter of course. For example, suppose that we are transmitting English-
language messages using a Caesar cipher with a shift of one (K = 1). A sends the
following legitimate ciphertext:

nbsftfbupbutboeepftfbupbutboemjuumfmbnctfbujwz

B decrypts to produce the following plaintext:

mareseatoatsanddoeseatoatsandlittlelambseativy

A simple frequency analysis confirms that this message has the profile of ordinary
English. On the other hand, if an opponent generates the following random se-
quence of letters:

zuvrsoevgqxlzwigamdvnmhpmccxiuureosfbcebtqxsxq

this decrypts to

ytuqrndufpwkyvhfzlcumlgolbbwhttqdnreabdaspwrwp

which does not fit the profile of ordinary English.

386 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

It may be difficult to determine automatically if incoming ciphertext de-
crypts to intelligible plaintext. If the plaintext is, say, a binary object file or digi-
tized X-rays, determination of properly formed and therefore authentic plaintext
may be difficult. Thus, an opponent could achieve a certain level of disruption
simply by issuing messages with random content purporting to come from a
legitimate user.

One solution to this problem is to force the plaintext to have some structure
that is easily recognized but that cannot be replicated without recourse to the en-
cryption function. We could, for example, append an error-detecting code, also
known as a frame check sequence (FCS) or checksum, to each message before en-
cryption, as illustrated in Figure 12.2a. A prepares a plaintext message M and then
provides this as input to a function F that produces an FCS. The FCS is appended to
M and the entire block is then encrypted. At the destination, B decrypts the incom-
ing block and treats the results as a message with an appended FCS. B applies the
same function F to attempt to reproduce the FCS. If the calculated FCS is equal to
the incoming FCS, then the message is considered authentic. It is unlikely that any
random sequence of bits would exhibit the desired relationship.

Note that the order in which the FCS and encryption functions are performed
is critical. The sequence illustrated in Figure 12.2a is referred to in [DIFF79] as
internal error control, which the authors contrast with external error control
(Figure 12.2b). With internal error control, authentication is provided because an
opponent would have difficulty generating ciphertext that, when decrypted, would
have valid error control bits. If instead the FCS is the outer code, an opponent can
construct messages with valid error-control codes. Although the opponent cannot
know what the decrypted plaintext will be, he or she can still hope to create confu-
sion and disrupt operations.

Figure 12.2 Internal and External Error Control

(b) External error control

Destination BSource A

K K

M | |

F

(a) Internal error control

MD
F

Compare

EM

F(M) F(M)
E(K, [M || F(M)])

M | |E
D

K

F

Compare

K

F

E(K, M)

F(E(K, M))

E(K, M)

M

12.2 / MESSAGE AUTHENTICATION FUNCTIONS 387

An error-control code is just one example; in fact, any sort of structuring
added to the transmitted message serves to strengthen the authentication capability.
Such structure is provided by the use of a communications architecture consisting
of layered protocols. As an example, consider the structure of messages transmit-
ted using the TCP/IP protocol architecture. Figure 12.3 shows the format of a TCP
segment, illustrating the TCP header. Now suppose that each pair of hosts shared
a unique secret key, so that all exchanges between a pair of hosts used the same
key, regardless of application. Then we could simply encrypt all of the datagram ex-
cept the IP header. Again, if an opponent substituted some arbitrary bit pattern for
the encrypted TCP segment, the resulting plaintext would not include a meaning-
ful header. In this case, the header includes not only a checksum (which covers the
header) but also other useful information, such as the sequence number. Because
successive TCP segments on a given connection are numbered sequentially, encryp-
tion assures that an opponent does not delay, misorder, or delete any segments.

PUBLIC-KEY ENCRYPTION The straightforward use of public-key encryption
(Figure 12.1b) provides confidentiality but not authentication. The source (A) uses
the public key PUb of the destination (B) to encrypt M. Because only B has the cor-
responding private key PRb, only B can decrypt the message. This scheme provides
no authentication, because any opponent could also use B’s public key to encrypt a
message and claim to be A.

To provide authentication, A uses its private key to encrypt the message, and
B uses A’s public key to decrypt (Figure 12.1c). This provides authentication using
the same type of reasoning as in the symmetric encryption case: The message must
have come from A because A is the only party that possesses PRa and therefore
the only party with the information necessary to construct ciphertext that can be
decrypted with PUa. Again, the same reasoning as before applies: There must be
some internal structure to the plaintext so that the receiver can distinguish between
well-formed plaintext and random bits.

Figure 12.3 TCP Segment

Source port Destination port

Checksum Urgent pointer

Sequence number

Acknowledgment number

Options + padding

Application data

Reserved Flags WindowDataoffset

0Bit: 4 10 16 31

20
o

ct
et

s

388 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

Assuming there is such structure, then the scheme of Figure 12.1c does pro-
vide authentication. It also provides what is known as digital signature.1 Only A
could have constructed the ciphertext because only A possesses PRa. Not even B,
the recipient, could have constructed the ciphertext. Therefore, if B is in possession
of the ciphertext, B has the means to prove that the message must have come from
A. In effect, A has “signed” the message by using its private key to encrypt. Note
that this scheme does not provide confidentiality. Anyone in possession of A’s pub-
lic key can decrypt the ciphertext.

To provide both confidentiality and authentication, A can encrypt M first
using its private key, which provides the digital signature, and then using B’s pub-
lic key, which provides confidentiality (Figure 12.1d). The disadvantage of this ap-
proach is that the public-key algorithm, which is complex, must be exercised four
times rather than two in each communication.

Message Authentication Code

An alternative authentication technique involves the use of a secret key to generate
a small fixed-size block of data, known as a cryptographic checksum or MAC, that is
appended to the message. This technique assumes that two communicating parties,
say A and B, share a common secret key K. When A has a message to send to B, it
calculates the MAC as a function of the message and the key:

MAC = C(K, M)

where

M = input message
C = MAC function
K = shared secret key
MAC = message authentication code

The message plus MAC are transmitted to the intended recipient. The recipient
performs the same calculation on the received message, using the same secret key,
to generate a new MAC. The received MAC is compared to the calculated MAC
(Figure 12.4a). If we assume that only the receiver and the sender know the identity
of the secret key, and if the received MAC matches the calculated MAC, then

1. The receiver is assured that the message has not been altered. If an attacker al-
ters the message but does not alter the MAC, then the receiver’s calculation of
the MAC will differ from the received MAC. Because the attacker is assumed
not to know the secret key, the attacker cannot alter the MAC to correspond
to the alterations in the message.

2. The receiver is assured that the message is from the alleged sender. Because
no one else knows the secret key, no one else could prepare a message with a
proper MAC.

1This is not the way in which digital signatures are constructed, as we shall see, but the principle is the
same.

Hiva-Network.Com

12.2 / MESSAGE AUTHENTICATION FUNCTIONS 389

3. If the message includes a sequence number (such as is used with HDLC, X.25,
and TCP), then the receiver can be assured of the proper sequence because an
attacker cannot successfully alter the sequence number.

A MAC function is similar to encryption. One difference is that the MAC
algorithm need not be reversible, as it must be for decryption. In general, the MAC
function is a many-to-one function. The domain of the function consists of messages
of some arbitrary length, whereas the range consists of all possible MACs and all
possible keys. If an n-bit MAC is used, then there are 2n possible MACs, whereas
there are N possible messages with N W 2n. Furthermore, with a k-bit key, there
are 2k possible keys.

For example, suppose that we are using 100-bit messages and a 10-bit MAC.
Then, there are a total of 2100 different messages but only 210 different MACs. So,
on average, each MAC value is generated by a total of 2100/210 = 290 different mes-
sages. If a 5-bit key is used, then there are 25 = 32 different mappings from the set
of messages to the set of MAC values.

It turns out that, because of the mathematical properties of the authentication
function, it is less vulnerable to being broken than encryption.

The process depicted in Figure 12.4a provides authentication but not confiden-
tiality, because the message as a whole is transmitted in the clear. Confidentiality
can be provided by performing message encryption either after (Figure 12.4b) or
before (Figure 12.4c) the MAC algorithm. In both these cases, two separate keys are

Figure 12.4 Basic Uses of Message Authentication code (MAC)

Destination BSource A

M | |

K

C

(a) Message authentication

M
E

| |

(c) Message authentication and confidentiality; authentication tied to ciphertext

M

C(K, M)

E(K2, [M || C(K1, M)])

C(K1, E(K2, M))

E(K2, M)

C

CompareK

EM | |

K1

K1

K2

K2

K2
K1

K1

K2

C

(b) Message authentication and confidentiality; authentication tied to plaintext

MD
C

Compare

C

C

Compare

D
M

C(K1, M)

390 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

needed, each of which is shared by the sender and the receiver. In the first case, the
MAC is calculated with the message as input and is then concatenated to the mes-
sage. The entire block is then encrypted. In the second case, the message is encrypted
first. Then the MAC is calculated using the resulting ciphertext and is concatenated
to the ciphertext to form the transmitted block. Typically, it is preferable to tie the
authentication directly to the plaintext, so the method of Figure 12.4b is used.

Because symmetric encryption will provide authentication and because it is
widely used with readily available products, why not simply use this instead of a
separate message authentication code? [DAVI89] suggests three situations in which
a message authentication code is used.

1. There are a number of applications in which the same message is broadcast to
a number of destinations. Examples are notification to users that the network
is now unavailable or an alarm signal in a military control center. It is cheaper
and more reliable to have only one destination responsible for monitoring au-
thenticity. Thus, the message must be broadcast in plaintext with an associated
message authentication code. The responsible system has the secret key and
performs authentication. If a violation occurs, the other destination systems
are alerted by a general alarm.

2. Another possible scenario is an exchange in which one side has a heavy load
and cannot afford the time to decrypt all incoming messages. Authentication is
carried out on a selective basis, messages being chosen at random for checking.

3. Authentication of a computer program in plaintext is an attractive service. The
computer program can be executed without having to decrypt it every time,
which would be wasteful of processor resources. However, if a message au-
thentication code were attached to the program, it could be checked whenever
assurance was required of the integrity of the program.

Three other rationales may be added.

4. For some applications, it may not be of concern to keep messages secret, but
it is important to authenticate messages. An example is the Simple Network
Management Protocol Version 3 (SNMPv3), which separates the functions of
confidentiality and authentication. For this application, it is usually important
for a managed system to authenticate incoming SNMP messages, particularly
if the message contains a command to change parameters at the managed sys-
tem. On the other hand, it may not be necessary to conceal the SNMP traffic.

5. Separation of authentication and confidentiality functions affords architec-
tural flexibility. For example, it may be desired to perform authentication at
the application level but to provide confidentiality at a lower level, such as the
transport layer.

6. A user may wish to prolong the period of protection beyond the time of recep-
tion and yet allow processing of message contents. With message encryption, the
protection is lost when the message is decrypted, so the message is protected
against fraudulent modifications only in transit but not within the target system.

Finally, note that the MAC does not provide a digital signature, because both
sender and receiver share the same key.

12.3 / REQUIREMENTS FOR MESSAGE AUTHENTICATION CODES 391

12.3 REQUIREMENTS FOR MESSAGE AUTHENTICATION CODES

A MAC, also known as a cryptographic checksum, is generated by a function C of
the form

T = MAC(K, M)

where M is a variable-length message, K is a secret key shared only by sender and re-
ceiver, and MAC(K, M) is the fixed-length authenticator, sometimes called a tag. The
tag is appended to the message at the source at a time when the message is assumed or
known to be correct. The receiver authenticates that message by recomputing the tag.

When an entire message is encrypted for confidentiality, using either symmet-
ric or asymmetric encryption, the security of the scheme generally depends on the
bit length of the key. Barring some weakness in the algorithm, the opponent must
resort to a brute-force attack using all possible keys. On average, such an attack will
require 2(k - 1) attempts for a k-bit key. In particular, for a ciphertext-only attack, the
opponent, given ciphertext C, performs Pi = D(Ki, C) for all possible key values Ki
until a Pi is produced that matches the form of acceptable plaintext.

In the case of a MAC, the considerations are entirely different. In general,
the MAC function is a many-to-one function, due to the many-to-one nature of
the function. Using brute-force methods, how would an opponent attempt to dis-
cover a key? If confidentiality is not employed, the opponent has access to plain-
text messages and their associated MACs. Suppose k 7 n; that is, suppose that
the key size is greater than the MAC size. Then, given a known M1 and T1, with
T1 = MAC(K, M1), the cryptanalyst can perform Ti = MAC(Ki, M1) for all pos-
sible key values ki. At least one key is guaranteed to produce a match of Ti = T1.
Note that a total of 2k tags will be produced, but there are only 2n 6 2k different tag
values. Thus, a number of keys will produce the correct tag and the opponent has no
way of knowing which is the correct key. On average, a total of 2k/2n = 2(k - n) keys
will produce a match. Thus, the opponent must iterate the attack.

■ Round 1

Given: M1, T1 = MAC(K, M1)
Compute Ti = MAC(Ki, M1) for all 2k keys
Number of matches L 2(k - n)

■ Round 2

Given: M2, T2 = MAC(K, M2)
Compute Ti = MAC(Ki, M2) for the 2(k - n) keys resulting from Round 1
Number of matches L 2(k - 2 * n)

And so on. On average, a rounds will be needed k = a * n. For example, if an
80-bit key is used and the tag is 32 bits, then the first round will produce about 248
possible keys. The second round will narrow the possible keys to about 216 possibili-
ties. The third round should produce only a single key, which must be the one used
by the sender.

392 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

If the key length is less than or equal to the tag length, then it is likely that a
first round will produce a single match. It is possible that more than one key will
produce such a match, in which case the opponent would need to perform the same
test on a new (message, tag) pair.

Thus, a brute-force attempt to discover the authentication key is no less ef-
fort and may be more effort than that required to discover a decryption key of the
same length. However, other attacks that do not require the discovery of the key
are possible.

Consider the following MAC algorithm. Let M = (X1 }X2 } c }Xm) be a
message that is treated as a concatenation of 64-bit blocks Xi. Then define

∆(M) = X1⊕ X2⊕ c ⊕ Xm
MAC(K, M) = E(K, ∆(M))

where ⊕ is the exclusive-OR (XOR) operation and the encryption algorithm
is DES in electronic codebook mode. Thus, the key length is 56 bits, and the tag
length is 64 bits. If an opponent observes {M }MAC(K, M)}, a brute-force attempt
to determine K will require at least 256 encryptions. But the opponent can attack the
system by replacing X1 through Xm - 1 with any desired values Y1 through Ym - 1 and
replacing Xm with Ym, where Ym is calculated as

Ym = Y1⊕ Y2⊕ g ⊕ Ym - 1⊕ ∆(M)

The opponent can now concatenate the new message, which consists of Y1
through Ym, using the original tag to form a message that will be accepted as authen-
tic by the receiver. With this tactic, any message of length 64 * (m - 1) bits can be
fraudulently inserted.

Thus, in assessing the security of a MAC function, we need to consider the
types of attacks that may be mounted against it. With that in mind, let us state the
requirements for the function. Assume that an opponent knows the MAC func-
tion but does not know K. Then the MAC function should satisfy the following
requirements.

1. If an opponent observes M and MAC(K, M), it should be computationally
infeasible for the opponent to construct a message M′ such that

MAC(K, M′) = MAC(K, M)

2. MAC(K, M) should be uniformly distributed in the sense that for randomly
chosen messages, M and M′, the probability that MAC(K, M) = MAC(K, M′)
is 2-n, where n is the number of bits in the tag.

3. Let M′ be equal to some known transformation on M. That is, M′ = f(M). For
example, f may involve inverting one or more specific bits. In that case,

Pr [MAC(K, M) = MAC(K, M′)] = 2-n

The first requirement speaks to the earlier example, in which an opponent is
able to construct a new message to match a given tag, even though the opponent
does not know and does not learn the key. The second requirement deals with the
need to thwart a brute-force attack based on chosen plaintext. That is, if we assume

12.4 / SECURITY OF MACs 393

that the opponent does not know K but does have access to the MAC function and
can present messages for MAC generation, then the opponent could try various
messages until finding one that matches a given tag. If the MAC function exhibits
uniform distribution, then a brute-force method would require, on average, 2(n - 1)
attempts before finding a message that fits a given tag.

The final requirement dictates that the authentication algorithm should not be
weaker with respect to certain parts or bits of the message than others. If this were
not the case, then an opponent who had M and MAC(K, M) could attempt varia-
tions on M at the known “weak spots” with a likelihood of early success at produc-
ing a new message that matched the old tags.

12.4 SECURITY OF MACs

Just as with encryption algorithms and hash functions, we can group attacks on
MACs into two categories: brute-force attacks and cryptanalysis.

Brute-Force Attacks

A brute-force attack on a MAC is a more difficult undertaking than a brute-force
attack on a hash function because it requires known message-tag pairs. Let us see
why this is so. To attack a hash code, we can proceed in the following way. Given
a fixed message x with n-bit hash code h = H(x), a brute-force method of finding
a collision is to pick a random bit string y and check if H(y) = H(x). The attacker
can do this repeatedly off line. Whether an off-line attack can be used on a MAC
algorithm depends on the relative size of the key and the tag.

To proceed, we need to state the desired security property of a MAC algo-
rithm, which can be expressed as follows.

■ Computation resistance: Given one or more text-MAC pairs [xi, MAC(K, xi)],
it is computationally infeasible to compute any text-MAC pair [x, MAC(K, x)]
for any new input x ≠ xi.

In other words, the attacker would like to come up with the valid MAC code for a
given message x. There are two lines of attack possible: attack the key space and at-
tack the MAC value. We examine each of these in turn.

If an attacker can determine the MAC key, then it is possible to generate a
valid MAC value for any input x. Suppose the key size is k bits and that the attacker
has one known text-tag pair. Then the attacker can compute the n-bit tag on the
known text for all possible keys. At least one key is guaranteed to produce the cor-
rect tag, namely, the valid key that was initially used to produce the known text-tag
pair. This phase of the attack takes a level of effort proportional to 2k (that is, one
operation for each of the 2k possible key values). However, as was described earlier,
because the MAC is a many-to-one mapping, there may be other keys that produce
the correct value. Thus, if more than one key is found to produce the correct value,
additional text-tag pairs must be tested. It can be shown that the level of effort
drops off rapidly with each additional text-MAC pair and that the overall level of
effort is roughly 2k [MENE97].

394 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

An attacker can also work on the tag without attempting to recover the key.
Here, the objective is to generate a valid tag for a given message or to find a message
that matches a given tag. In either case, the level of effort is comparable to that for
attacking the one-way or weak collision-resistant property of a hash code, or 2n.
In the case of the MAC, the attack cannot be conducted off line without further
input; the attacker will require chosen text-tag pairs or knowledge of the key.

To summarize, the level of effort for brute-force attack on a MAC algorithm
can be expressed as min(2k, 2n). The assessment of strength is similar to that for
symmetric encryption algorithms. It would appear reasonable to require that the
key length and tag length satisfy a relationship such as min(k, n) Ú N, where N is
perhaps in the range of 128 bits.

Cryptanalysis

As with encryption algorithms and hash functions, cryptanalytic attacks on MAC
algorithms seek to exploit some property of the algorithm to perform some attack
other than an exhaustive search. The way to measure the resistance of a MAC algo-
rithm to cryptanalysis is to compare its strength to the effort required for a brute-
force attack. That is, an ideal MAC algorithm will require a cryptanalytic effort
greater than or equal to the brute-force effort.

There is much more variety in the structure of MACs than in hash functions,
so it is difficult to generalize about the cryptanalysis of MACs. Furthermore, far less
work has been done on developing such attacks. A useful survey of some methods
for specific MACs is [PREN96].

12.5 MACs BASED ON HASH FUNCTIONS: HMAC

Later in this chapter, we look at examples of a MAC based on the use of a symmetric
block cipher. This has traditionally been the most common approach to constructing
a MAC. In recent years, there has been increased interest in developing a MAC de-
rived from a cryptographic hash function. The motivations for this interest are

1. Cryptographic hash functions such as MD5 and SHA generally execute faster
in software than symmetric block ciphers such as DES.

2. Library code for cryptographic hash functions is widely available.

With the development of AES and the more widespread availability of code
for encryption algorithms, these considerations are less significant, but hash-based
MACs continue to be widely used.

A hash function such as SHA was not designed for use as a MAC and can-
not be used directly for that purpose, because it does not rely on a secret key.
There have been a number of proposals for the incorporation of a secret key into
an existing hash algorithm. The approach that has received the most support is
HMAC [BELL96a, BELL96b]. HMAC has been issued as RFC 2104, has been
chosen as the mandatory-to-implement MAC for IP security, and is used in other
Internet protocols, such as SSL. HMAC has also been issued as a NIST standard
(FIPS 198).

12.5 / MACs BASED ON HASH FUNCTIONS: HMAC 395

HMAC Design Objectives

RFC 2104 lists the following design objectives for HMAC.

■ To use, without modifications, available hash functions. In particular, to use
hash functions that perform well in software and for which code is freely and
widely available.

■ To allow for easy replaceability of the embedded hash function in case faster
or more secure hash functions are found or required.

■ To preserve the original performance of the hash function without incurring a
significant degradation.

■ To use and handle keys in a simple way.

■ To have a well understood cryptographic analysis of the strength of the au-
thentication mechanism based on reasonable assumptions about the embed-
ded hash function.

The first two objectives are important to the acceptability of HMAC. HMAC
treats the hash function as a “black box.” This has two benefits. First, an existing im-
plementation of a hash function can be used as a module in implementing HMAC.
In this way, the bulk of the HMAC code is prepackaged and ready to use without
modification. Second, if it is ever desired to replace a given hash function in an
HMAC implementation, all that is required is to remove the existing hash function
module and drop in the new module. This could be done if a faster hash function
were desired. More important, if the security of the embedded hash function were
compromised, the security of HMAC could be retained simply by replacing the em-
bedded hash function with a more secure one (e.g., replacing SHA-2 with SHA-3).

The last design objective in the preceding list is, in fact, the main advantage
of HMAC over other proposed hash-based schemes. HMAC can be proven secure
provided that the embedded hash function has some reasonable cryptographic
strengths. We return to this point later in this section, but first we examine the struc-
ture of HMAC.

HMAC Algorithm

Figure 12.5 illustrates the overall operation of HMAC. Define the following terms.

H = embedded hash function (e.g., MD5, SHA-1, RIPEMD-160)
IV = initial value input to hash function
M = message input to HMAC (including the padding specified in the embedded

hash function)
Yi = i th block of M, 0 … i … (L - 1)
L = number of blocks in M
b = number of bits in a block
n = length of hash code produced by embedded hash function
K = secret key; recommended length is Ú n; if key length is greater than b, the

key is input to the hash function to produce an n-bit key
K+ = K padded with zeros on the left so that the result is b bits in length

396 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

ipad = 00110110 (36 in hexadecimal) repeated b/8 times
opad = 01011100 (5C in hexadecimal) repeated b/8 times

Then HMAC can be expressed as

HMAC(K, M) = H[(K+ ⊕ opad) }H[(K+ ⊕ ipad) }M]]

We can describe the algorithm as follows.

1. Append zeros to the left end of K to create a b-bit string K+ (e.g., if K is of
length 160 bits and b = 512, then K will be appended with 44 zeroes).

2. XOR (bitwise exclusive-OR) K+ with ipad to produce the b-bit block Si.

3. Append M to Si.

4. Apply H to the stream generated in step 3.

5. XOR K+ with opad to produce the b-bit block So.

6. Append the hash result from step 4 to So.

7. Apply H to the stream generated in step 6 and output the result.

Note that the XOR with ipad results in flipping one-half of the bits of K.
Similarly, the XOR with opad results in flipping one-half of the bits of K, using a

Figure 12.5 HMAC Structure

K+

Si

So

Y0 Y1 YL–1

b bits

b bits

b bits b bits

ipad

K+ opad

HashIV n bits

n bits

Pad to b bits

HashIV n bits

n bits

HMAC(K, M)

H(Si || M)

12.5 / MACs BASED ON HASH FUNCTIONS: HMAC 397

different set of bits. In effect, by passing Si and So through the compression function
of the hash algorithm, we have pseudorandomly generated two keys from K.

HMAC should execute in approximately the same time as the embedded hash
function for long messages. HMAC adds three executions of the hash compression
function (for Si, So, and the block produced from the inner hash).

A more efficient implementation is possible, as shown in Figure 12.6. Two
quantities are precomputed:

f(IV, (K+ ⊕ ipad))
f(IV, (K+ ⊕ opad))

where f(cv, block) is the compression function for the hash function, which takes as
arguments a chaining variable of n bits and a block of b bits and produces a chain-
ing variable of n bits. These quantities only need to be computed initially and every
time the key changes. In effect, the precomputed quantities substitute for the initial
value (IV) in the hash function. With this implementation, only one additional in-
stance of the compression function is added to the processing normally produced

Figure 12.6 Efficient Implementation of HMAC

b bits b bits b bits

Precomputed Computed per message

HashIV n bits

b bits

n bits

Pad to b bits

n bits

n bits

HMAC(K, M)

f

IV

b bits

f f

K+

Si

So

Y0 Y1

ipad

K+ opad

YL–1

H(Si || M)

Hiva-Network.Com

398 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

by the hash function. This more efficient implementation is especially worthwhile if
most of the messages for which a MAC is computed are short.

Security of HMAC

The security of any MAC function based on an embedded hash function depends
in some way on the cryptographic strength of the underlying hash function. The
appeal of HMAC is that its designers have been able to prove an exact relation-
ship between the strength of the embedded hash function and the strength of
HMAC.

The security of a MAC function is generally expressed in terms of the prob-
ability of successful forgery with a given amount of time spent by the forger and
a given number of message-tag pairs created with the same key. In essence, it is
proved in [BELL96a] that for a given level of effort (time, message–tag pairs) on
messages generated by a legitimate user and seen by the attacker, the probability
of successful attack on HMAC is equivalent to one of the following attacks on the
embedded hash function.

1. The attacker is able to compute an output of the compression function even
with an IV that is random, secret, and unknown to the attacker.

2. The attacker finds collisions in the hash function even when the IV is random
and secret.

In the first attack, we can view the compression function as equivalent to the
hash function applied to a message consisting of a single b-bit block. For this attack,
the IV of the hash function is replaced by a secret, random value of n bits. An attack
on this hash function requires either a brute-force attack on the key, which is a level
of effort on the order of 2n, or a birthday attack, which is a special case of the second
attack, discussed next.

In the second attack, the attacker is looking for two messages M and M′ that
produce the same hash: H(M) = H(M′). This is the birthday attack discussed in
Chapter 11. We have shown that this requires a level of effort of 2n/2 for a hash
length of n. On this basis, the security of MD5 is called into question, because a
level of effort of 264 looks feasible with today’s technology. Does this mean that
a 128-bit hash function such as MD5 is unsuitable for HMAC? The answer is no,
because of the following argument. To attack MD5, the attacker can choose any
set of messages and work on these off line on a dedicated computing facility to
find a collision. Because the attacker knows the hash algorithm and the default IV,
the attacker can generate the hash code for each of the messages that the attacker
generates. However, when attacking HMAC, the attacker cannot generate mes-
sage/code pairs off line because the attacker does not know K. Therefore, the at-
tacker must observe a sequence of messages generated by HMAC under the same
key and perform the attack on these known messages. For a hash code length of
128 bits, this requires 264 observed blocks (272 bits) generated using the same key.
On a 1-Gbps link, one would need to observe a continuous stream of messages
with no change in key for about 150,000 years in order to succeed. Thus, if speed
is a concern, it is fully acceptable to use MD5 rather than SHA-1 as the embedded
hash function for HMAC.

12.6 / MACs BASED ON BLOCK CIPHERS: DAA AND CMAC 399

12.6 MACs BASED ON BLOCK CIPHERS: DAA AND CMAC

In this section, we look at two MACs that are based on the use of a block cipher
mode of operation. We begin with an older algorithm, the Data Authentication
Algorithm (DAA), which is now obsolete. Then we examine CMAC, which is de-
signed to overcome the deficiencies of DAA.

Data Authentication Algorithm

The Data Authentication Algorithm (DAA), based on DES, has been one of the
most widely used MACs for a number of years. The algorithm is both a FIPS pub-
lication (FIPS PUB 113) and an ANSI standard (X9.17). However, as we discuss
subsequently, security weaknesses in this algorithm have been discovered, and it is
being replaced by newer and stronger algorithms.

The algorithm can be defined as using the cipher block chaining (CBC) mode
of operation of DES (Figure 6.4) with an initialization vector of zero. The data (e.g.,
message, record, file, or program) to be authenticated are grouped into contiguous
64-bit blocks: D1, D2, c , DN. If necessary, the final block is padded on the right
with zeroes to form a full 64-bit block. Using the DES encryption algorithm E and a
secret key K, a data authentication code (DAC) is calculated as follows (Figure 12.7).

O1 = E(K, D)
O2 = E(K, [D2⊕O1])
O3 = E(K, [D3⊕O2])#
#
#
ON = E(K, [DN⊕ON - 1])

Figure 12.7 Data Authentication Algorithm (FIPS PUB 113)

Time = 1

DES
encrypt

K
(56 bits)

Time = 2

K

+ + +

K K

Time = NTime = N – 1

O1
(64 bits)

O2

D1
(64 bits) D2 DN–1

ON

DN

ON–1

DAC
(16 to 64 bits)

DES
encrypt

DES
encrypt

DES
encrypt

400 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

The DAC consists of either the entire block ON or the leftmost M bits of the
block, with 16 … M … 64.

Cipher-Based Message Authentication Code (CMAC)

As was mentioned, DAA has been widely adopted in government and industry.
[BELL00] demonstrated that this MAC is secure under a reasonable set of security
criteria, with the following restriction. Only messages of one fixed length of mn bits
are processed, where n is the cipher block size and m is a fixed positive integer. As
a simple example, notice that given the CBC MAC of a one-block message X, say
T = MAC(K, X), the adversary immediately knows the CBC MAC for the two-
block message X } (X⊕ T) since this is once again T.

Black and Rogaway [BLAC00] demonstrated that this limitation could be
overcome using three keys: one key K of length k to be used at each step of the
cipher block chaining and two keys of length b, where b is the cipher block length.
This proposed construction was refined by Iwata and Kurosawa so that the two
n-bit keys could be derived from the encryption key, rather than being provided
separately [IWAT03]. This refinement, adopted by NIST, is the Cipher-based
Message Authentication Code (CMAC) mode of operation for use with AES and
triple DES. It is specified in NIST Special Publication 800-38B.

First, let us define the operation of CMAC when the message is an integer
multiple n of the cipher block length b. For AES, b = 128, and for triple DES,
b = 64. The message is divided into n blocks (M1, M2, c , Mn). The algorithm
makes use of a k-bit encryption key K and a b-bit constant, K1. For AES, the key
size k is 128, 192, or 256 bits; for triple DES, the key size is 112 or 168 bits. CMAC is
calculated as follows (Figure 12.8).

C1 = E(K, M1)
C2 = E(K, [M2⊕ C1])
C3 = E(K, [M3⊕ C2])#
#
#

Cn = E(K, [Mn⊕ Cn - 1 ⊕ K1])
T = MSBTlen(Cn)

where

T = message authentication code, also referred to as the tag
Tlen = bit length of T
MSBs(X) = the s leftmost bits of the bit string X

If the message is not an integer multiple of the cipher block length, then the
final block is padded to the right (least significant bits) with a 1 and as many 0s as
necessary so that the final block is also of length b. The CMAC operation then pro-
ceeds as before, except that a different b-bit key K2 is used instead of K1.

12.6 / MACs BASED ON HASH FUNCTIONS: HMAC 401

The two b-bit keys are derived from the k-bit encryption key as follows.

L = E(K, 0b)
K1 = L # x
K2 = L # x2 = (L # x) # x

where multiplication ( # ) is done in the finite field GF(2b) and x and x2 are first- and
second-order polynomials that are elements of GF(2b). Thus, the binary represen-
tation of x consists of b - 2 zeros followed by 10; the binary representation of x2
consists of b - 3 zeros followed by 100. The finite field is defined with respect to
an irreducible polynomial that is lexicographically first among all such polynomials
with the minimum possible number of nonzero terms. For the two approved block
sizes, the polynomials are x64 + x4 + x3 + x + 1 and x128 + x7 + x2 + x + 1.

To generate K1 and K2, the block cipher is applied to the block that consists
entirely of 0 bits. The first subkey is derived from the resulting ciphertext by a
left shift of one bit and, conditionally, by XORing a constant that depends on the
block size. The second subkey is derived in the same manner from the first subkey.
This property of finite fields of the form GF(2b) was explained in the discussion of
MixColumns in Chapter 6.

Figure 12.8 Cipher-Based Message Authentication Code (CMAC)

EncryptK K K

T

Encrypt Encrypt

MSB(Tlen)

M1

K1

K2

M2 Mn

(a) Message length is integer multiple of block size

EncryptK K K

T

Encrypt Encrypt

MSB(Tlen)

10...0

(b) Message length is not integer multiple of block size

b

k

MnM1 M2

402 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

12.7 AUTHENTICATED ENCRYPTION: CCM AND GCM

Authenticated encryption (AE) is a term used to describe encryption systems that
simultaneously protect confidentiality and authenticity (integrity) of communica-
tions. Many applications and protocols require both forms of security, but until re-
cently the two services have been designed separately.

There are four common approaches to providing both confidentiality and en-
cryption for a message M.

■ Hashing followed by encryption (H S E): First compute the cryptographic
hash function over M as h = H(M). Then encrypt the message plus hash func-
tion: E(K, (M }h)).

■ Authentication followed by encryption (A S E): Use two keys. First authen-
ticate the plaintext by computing the MAC value as T = MAC(K1, M). Then
encrypt the message plus tag: E(K2, [M }T ]). This approach is taken by the
SSL/TLS protocols (Chapter 17).

■ Encryption followed by authentication (E S A): Use two keys. First encrypt
the message to yield the ciphertext C = E(K2, M). Then authenticate the
ciphertext with T = MAC(K1, C) to yield the pair (C, T). This approach is
used in the IPSec protocol (Chapter 20).

■ Independently encrypt and authenticate (E + A). Use two keys. Encrypt
the message to yield the ciphertext C = E(K2, M). Authenticate the plain-
text with T = MAC(K1, M) to yield the pair (C, T). These operations can
be performed in either order. This approach is used by the SSH protocol
(Chapter 17).

Both decryption and verification are straightforward for each approach. For
H S E, A S E, and E + A, decrypt first, then verify. For E S A, verify first, then
decrypt. There are security vulnerabilities with all of these approaches. The H S E
approach is used in the Wired Equivalent Privacy (WEP) protocol to protect WiFi
networks. This approach had fundamental weaknesses and led to the replacement of
the WEP protocol. [BLAC05] and [BELL00] point out that there are security con-
cerns in each of the three encryption/MAC approaches listed above. Nevertheless,
with proper design, any of these approaches can provide a high level of security.
This is the goal of the two approaches discussed in this section, both of which have
been standardized by NIST.

Counter with Cipher Block Chaining-Message
Authentication Code

The CCM mode of operation was standardized by NIST specifically to sup-
port the security requirements of IEEE 802.11 WiFi wireless local area networks
(Chapter 18), but can be used in any networking application requiring authenti-
cated encryption. CCM is a variation of the encrypt-and-MAC approach to authen-
ticated encryption. It is defined in NIST SP 800-38C.

The key algorithmic ingredients of CCM are the AES encryption algorithm
(Chapter 6), the CTR mode of operation (Chapter 7), and the CMAC authentication

12.7 / AUTHENTICATED ENCRYPTION: CCM AND GCM 403

algorithm (Section 12.6). A single key K is used for both encryption and MAC algo-
rithms. The input to the CCM encryption process consists of three elements.

1. Data that will be both authenticated and encrypted. This is the plaintext mes-
sage P of data block.

2. Associated data A that will be authenticated but not encrypted. An example
is a protocol header that must be transmitted in the clear for proper protocol
operation but which needs to be authenticated.

3. A nonce N that is assigned to the payload and the associated data. This is a
unique value that is different for every instance during the lifetime of a pro-
tocol association and is intended to prevent replay attacks and certain other
types of attacks.

Figure 12.9 illustrates the operation of CCM. For authentication, the input
includes the nonce, the associated data, and the plaintext. This input is formatted
as a sequence of blocks B0 through Br. The first block contains the nonce plus some
formatting bits that indicate the lengths of the N, A, and P elements. This is fol-
lowed by zero or more blocks that contain A, followed by zero of more blocks that
contain P. The resulting sequence of blocks serves as input to the CMAC algorithm,
which produces a MAC value with length Tlen, which is less than or equal to the
block length (Figure 12.9a).

For encryption, a sequence of counters is generated that must be independent
of the nonce. The authentication tag is encrypted in CTR mode using the single
counter Ctr0. The Tlen most significant bits of the output are XORed with the tag to
produce an encrypted tag. The remaining counters are used for the CTR mode en-
cryption of the plaintext (Figure 7.7). The encrypted plaintext is concatenated with
the encrypted tag to form the ciphertext output (Figure 12.9b).

SP 800-38C defines the authentication/encryption process as follows.

1. Apply the formatting function to (N, A, P) to produce the blocks B0, B1, c , Br.
2. Set Y0 = E(K, B0).
3. For i = 1 to r, do Yi = E(K, (Bi⊕ Yi- 1)).
4. Set T = MSBTlen(Yr).
5. Apply the counter generation function to generate the counter blocks

Ctr0, Ctr1, c , Ctrm, where m = <Plen/128= .
6. For j = 0 to m, do Sj = E(K, Ctrj).
7. Set S = S1 }S2 } g }Sm.
8. Return C = (P⊕MSBPlen(S)) } (T⊕MSBTlen(S0)).

For decryption and verification, the recipient requires the following input: the
ciphertext C, the nonce N, the associated data A, the key K, and the initial counter
Ctr0. The steps are as follows.

1. If Clen … Tlen, then return INVALID.
2. Apply the counter generation function to generate the counter blocks

Ctr0, Ctr1, c , Ctrm, where m = <Clen/128= .
3. For j = 0 to m, do Sj = E(K, Ctrj).

404 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

4. Set S = S1 }S2 } g }Sm.
5. Set P = MSBClen - Tlen(C)⊕MSBClen - Tlen(S).
6. Set T = LSBTlen(C)⊕MSBTlen(S0).
7. Apply the formatting function to N, A, P) to produce the blocks B0, B1, c , Br.
8. Set Y0 = E(K, B0).
9. For i = 1 to r do Yi = E(K, (Bi⊕ Yi- 1)).

10. If T ≠ MSBTlen(Yr), then return INVALID, else return P.

Figure 12.9 Counter with Cipher Block Chaining-Message Authentication Code (CCM)

(a) Authentication

(b) Encryption

B0

Ctr0

B1 B2 Br

Tag

Tag

Nonce Plaintext

Plaintext

Ciphertext

Ass. Data

K CMAC

MSB(Tlen)
K

CTRCtr1, Ctr2, ..., Ctrm

EncryptK

12.7 / AUTHENTICATED ENCRYPTION: CCM AND GCM 405

CCM is a relatively complex algorithm. Note that it requires two complete
passes through the plaintext, once to generate the MAC value, and once for encryp-
tion. Further, the details of the specification require a tradeoff between the length
of the nonce and the length of the tag, which is an unnecessary restriction. Also note
that the encryption key is used twice with the CTR encryption mode: once to gener-
ate the tag and once to encrypt the plaintext plus tag. Whether these complexities
add to the security of the algorithm is not clear. In any case, two analyses of the
algorithm ([JONS02] and [ROGA03]) conclude that CCM provides a high level of
security.

Galois/Counter Mode

The GCM mode of operation, standardized by NIST in NIST SP 800-38D, is de-
signed to be parallelizable so that it can provide high throughput with low cost and
low latency. In essence, the message is encrypted in variant of CTR mode. The re-
sulting ciphertext is multiplied with key material and message length information
over GF(2128) to generate the authenticator tag. The standard also specifies a mode
of operation that supplies the MAC only, known as GMAC.

The GCM mode makes use of two functions: GHASH, which is a keyed hash
function, and GCTR, which is essentially the CTR mode with the counters deter-
mined by a simple increment by one operation.

GHASHH(X) takes a input the hash key H and a bit string X such that
len(X) = 128m bits for some positive integer m and produces a 128-bit MAC value.
The function may be specified as follows (Figure 12.10a).

1. Let X1, X2, c , Xm - 1, Xm denote the unique sequence of blocks such that
X = X1 }X2 } g }Xm - 1 }Xm.

2. Let Y0 be a block of 128 zeros, designated as 0
128.

3. For i = 1, c , m, let Yi = (Yi- 1⊕ Xi) # H, where # designates multiplication
in GF(2128).

4. Return Ym.

The GHASHH(X) function can be expressed as

(X1 # Hm)⊕ (X2 # Hm - 1)⊕ g ⊕ (Xm - 1 # H2)⊕ (Xm # H)
This formulation has desirable performance implications. If the same hash key

is to be used to authenticate multiple messages, then the values H2, H3, c can be
precalculated one time for use with each message to be authenticated. Then, the
blocks of the data to be authenticated (X1, X2, c , Xm) can be processed in paral-
lel, because the computations are independent of one another.

GCTRK(ICB, X) takes a input a secret key K and a bit string X arbitrary
length and returns a ciphertext Y of bit length (X). The function may be specified as
follows (Figure 12.10b).

1. If X is the empty string, then return the empty string as Y.

2. Let n = <(len(X)/128)= . That is, n is the smallest integer greater than or equal
to (X)/128.

406 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

3. Let X1, X2, c , Xn - 1, Xn* denote the unique sequence of bit strings such that

X = X1 }X2 } g }Xn - 1 }Xn*;
X1, X2, c , Xn - 1 are complete [email protected] blocks.

4. Let CB1 = ICB.
5. For, i = 2 to n let CBi = inc32(CBi - 1), where the inc32(S) function increments

the rightmost 32 bits of S by 1 mod 232, and the remaining bits are unchanged.

6. For i = 1 to n - 1, do Yi = Xi⊕ E(K, CBi).
7. Let Y n

* = Xn*⊕MSBlen(Xn*)(E(K, CBn)).
8. Let Y = Y1 }Y2 } c }Yn - 1 }Y n*

9. Return Y.

Note that the counter values can be quickly generated and that the encryption
operations can be performed in parallel.

Figure 12.10 GCM Authentication and Encryption Functions

(a) GHASHH(X1 || X2 || . . . || Xm) = Ym

X1

X1

X2

ICB

Xm

Y1

Y1

Y2 Ym

H

E

inc

H H

K

X2

CB2

Y2

EK

Xn–1

CBn–1

Yn–1

E

inc

K

Xn

CBn

Yn

E

MSB

K

*

(b) GCTRK(ICB, X1 || X2 || . . . || Xn) = Y1 || Y2 || . . . ||Yn**

*

Hiva-Network.Com

12.7 / AUTHENTICATED ENCRYPTION: CCM AND GCM 407

We can now define the overall authenticated encryption function
(Figure 12.11). The input consists of a secret key K, an initialization vector IV, a
plaintext P, and additional authenticated data A. The notation [x]s means the s-bit
binary representation of the nonnegative integer x. The steps are as follows.

1. Let H = E(K, 0128).
2. Define a block, J0, as

If len(IV) = 96, then let J0 = IV }031 }1.
If len (IV) ≠ 96, then let s = 128<len(IV)/128= - len(IV), and let
J0 = GHASHH(IV }0s + 64 } [len(IV)]64).

3. Let C = GCTRK(inc32(J0), P).
4. Let u = 128<len(C)/128= - len(C) and let v = 128<len(A)/128= - len(A).
5. Define a block, S, as

S = GHASHH(A }0v }C }0u } [len(A)]64 } [len(C)]64)

6. Let T = MSBt(GCTRK(J0, S)), where t is the supported tag length.
7. Return (C, T).

Figure 12.11 Galois Counter—Message Authentication Code (GCM)

IV

J0

J0

Plaintext

K

K

GCTR

encode

incr

GCTR

Tag

GHASH

A = Ass. Data C = Ciphertext [len(A)]64 [len(C )]640v 0u

MSBt

EK

H

0

408 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

In step 1, the hash key is generated by encrypting a block of all zeros with
the secret key K. In step 2, the pre-counter block (J0) is generated from the IV.
In particular, when the length of the IV is 96 bits, then the padding string 031 }1
is appended to the IV to form the pre-counter block. Otherwise, the IV is padded
with the minimum number of 0 bits, possibly none, so that the length of the result-
ing string is a multiple of 128 bits (the block size); this string in turn is appended
with 64 additional 0 bits, followed by the 64-bit representation of the length of
the IV, and the GHASH function is applied to the resulting string to form the
pre-counter block.

Thus, GCM is based on the CTR mode of operation and adds a MAC that au-
thenticates both the message and additional data that requires only authentication.
The function that computes the hash uses only multiplication in a Galois field. This
choice was made because the operation of multiplication is easy to perform within a
Galois field and is easily implemented in hardware [MCGR05].

[MCGR04] examines the available block cipher modes of operation and
shows that a CTR-based authenticated encryption approach is the most efficient
mode of operation for high-speed packet networks. The paper further demonstrates
that GCM meets a high level of security requirements.

12.8 KEY WRAPPING

Background

The most recent block cipher mode of operation defined by NIST is the Key Wrap
(KW) mode of operation (SP 800-38F), which uses AES or triple DEA as the un-
derlying encryption algorithm. The AES version is also documented in RFC 3394.

The purpose of key wrapping is to securely exchange a symmetric key to be
shared by two parties, using a symmetric key already shared by those parties. The
latter key is called a key encryption key (KEK).

Two questions need to be addressed at this point. First, why do we need to
use a symmetric key already known to two parties to encrypt a new symmetric key?
Such a requirement is found in a number of protocols described in this book, such
as the key management portion of IEEE 802.11 and IPsec. Quite often, a protocol
calls for a hierarchy of keys, with keys lower on the hierarchy used more frequently,
and changed more frequently to thwart attacks. A higher-level key, which is used in-
frequently and therefore more resistant to cryptanalysis, is used to encrypt a newly
created lower-level key so that it can be exchanged between parties that share the
higher-level key.

The second question is, why do we need a new mode? The intent of the new
mode is to operate on keys whose length is greater than the block size of the encryp-
tion algorithm. For example, AES uses a block size of 128 bits but can use a key
size of 128, 192, or 256 bits. In the latter two cases, encryption of the key involves
multiple blocks. We consider the value of key data to be greater than the value of
other data, because the key will be used multiple times, and compromise of the
key compromises all of the data encrypted with the key. Therefore, NIST desired

12.8 / KEY WRAPPING 409

a robust encryption mode. KW is robust in the sense that each bit of output can be
expected to depend in a nontrivial fashion on each bit of input. This is not the case
for any of the other modes of operation that we have described. For example, in
all of the modes so far described, the last block of plaintext only influences the last
block of ciphertext. Similarly, the first block of ciphertext is derived only from the
first block of plaintext.

To achieve this robust operation, KW achieves a considerably lower through-
put than the other modes, but the tradeoff may be appropriate for some key
management applications. Also, KW is only used for small amounts of plaintext
compared to, say, the encryption of a message or a file.

The Key Wrapping Algorithm

The key wrapping algorithm operates on blocks of 64 bits. The input to the algo-
rithm consists of a 64-bit constant, discussed subsequently, and a plaintext key that
is divided into blocks of 64 bits. We use the following notation:

MSB64(W) most significant 64 bits of W

LSB64(W) least significant 64 bits of W

W temporary value; output of encryption function

bitwise exclusive-OR

} concatenation

K key encryption key

n number of 64-bit key data blocks

s number of stages in the wrapping process; s = 6n
Pi ith plaintext key data block; 1 … i … n
Ci ith ciphertext data block; 0 … i … n
A(t) 64-bit integrity check register after encryption stage t; 1 … t … s
A(0) initial integrity check value (ICV); in hexadecimal:

A6A6A6A6A6A6A6A6

R(t, i) 64-bit register i after encryption stage t; 1 … t … s; 1 … i … n

We now describe the key wrapping algorithm:

Inputs: Plaintext, n 64-bit values (P1, P2, c , Pn)
Key encryption key, K

Outputs: Ciphertext, (n + 1) 64-bit values (C0, C1, c , Cn)

1. Initialize variables.

A(0) = A6A6A6A6A6A6A6A6

for i = 1 to n

R(0, i) = Pi

⊕

410 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

2. Calculate intermediate values.

for t = 1 to s

W = E(K, [A(t−1) } R(t−1, 1)])
A(t) = t⊕ MSB64(W)
R(t, n) = LSB64(W)

for i = 1 to n−1

R(t, i) = R(t−1, i+1)

3. Output results.

C0 = A(s)

for i = 1 to n

Ci = R(s, i)

Note that the ciphertext is one block longer than the plaintext key, to ac-
commodate the ICV. Upon unwrapping (decryption), both the 64-bit ICV and the
plaintext key are recovered. If the recovered ICV differs from the input value of
hexadecimal A6A6A6A6A6A6A6A6, then an error or alteration has been detected
and the plaintext key is rejected. Thus, the key wrap algorithm provides not only
confidentiality but also data integrity.

Figure 12.12 illustrated the key wrapping algorithm for encrypting a 256-bit
key. Each box represents one encryption stage (one value of t). Note that the A
output is fed as input to the next stage (t + 1), whereas the R output skips forward
n stages (t + n), which in this example is n = 4. This arrangement further increases
the avalanche effect and the mixing of bits. To achieve this skipping of stages, a slid-
ing buffer is used, so that the R output from stage t is shifted in the buffer one posi-
tion for each stage, until it becomes the input for stage t + n. This might be clearer
if we expand the inner for loop for a 256-bit key (n = 4). Then the assignments are
as follows:

R(t, 1) = R(t - 1, 2)
R(t, 2) = R(t - 1, 3)
R(t, 3) = R(t - 1, 4)

For example, consider that at stage 5, the R output has a value of R(5, 4) = x.
At stage 6, we execute R(6, 3) = R(5, 4) = x. At stage 7, we execute R(7, 2) = R
(6, 3) = x. At stage 8, we execute R(8, 1) = R(7, 2) = x. So, at stage 9, the input
value of R(t - 1, 1) is R(8, 1) = x.

Figure 12.13 depicts the operation of stage t for a 256-bit key. The dashed
feedback lines indicate the assignment of new values to the stage variables.

Key Unwrapping

The key unwrapping algorithm can be defined as follows:

Inputs: Ciphertext, (n + 1) 64-bit values (C0, C1, c , Cn)
Key encryption key, K

Outputs: Plaintext, n 64-bit values (P1, P2, c , Pn), ICV

12.8 / KEY WRAPPING 411

1. Initialize variables.

A(s) = C0
for i = 1 to n

R(s, i) = Ci

2. Calculate intermediate values.

for t = s to 1

W = D(K, [(A(t)⊕ t) } R(t, n)])

Figure 12.12 Key Wrapping Operation for 256-Bit Key

t = 1 t = 2 t = 3 t = 4

t = 5 t = 6 t = 7 t = 8

t = 9 t = 10 t = 11 t = 12

t = 13 t = 14 t = 15 t = 16

t = 17 t = 18 t = 19 t = 20

t = 21

C0 = A(24)

A(0) A(1)

C1 = R(24, 1)
= R(21, 4)

P1 =
R(0, 1)

P2 =
R(0, 2)

P3 =
R(0, 3)

P4 =
R(0, 4) A(2) A(3)

C2 = R(24, 2)
= R(22, 4)

C3 = R(24, 3)
= R(23, 4)

C4 = R(24, 4)

t = 22 t = 23 t = 24

412 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

A(t–1) = MSB64(W)

R(t–1, 1) = LSB64(W)

for i = 2 to n

R(t–1, i) = R(t, i–1)

3. Output results.

if A(0) = A6A6A6A6A6A6A6A6

then

for i = 1 to n

P(i) = R(0, i)

else

return error

Note that the decryption function is used in the unwrapping algorithm.
We now demonstrate that the unwrap function is the inverse of the wrap func-

tion, that is, that the unwrap function recovers the plaintext key and the ICV. First,
note that because the index variable t is counted down from s to 1 for unwrapping,
stage t of the unwrap algorithm corresponds to stage t of the wrap algorithm. The
input variables to stage t of the wrap algorithm are indexed at t - 1 and the output
variables of stage t of the unwrap algorithm are indexed at t - 1. Thus, to demon-
strate that the two algorithms are inverses of each other, we need only demonstrate
that the output variables of stage t of the unwrap algorithm are equal to the input
variables to stage t of the wrap algorithm.

This demonstration is in two parts. First we demonstrate that the calculation
of A and R variables prior to the for loop are inverses. To do this, let us simplify
the notation a bit. Define the 128-bit value T to be the 64-bit value t followed by 64
zeros. Then, the first three lines of step 2 of the wrap algorithm can be written as the
following single line:

A(t) }R(t, n) = T⊕ E(K, [A(t - 1) }R(t - 1, 1)]) (12.1)

The first three lines of step 2 of the unwrap algorithm can be written as:

A(t - 1) }R(t - 1, 1) = D(K, ([A(t) }R(t, n)]⊕ T)) (12.2)

Figure 12.13 Key Wrapping Operation for 256-Bit Key: Stage t

A(t – 1)

Encrypt

MSB

K

t LSB

R(t – 1, 1) R(t – 1, 2)

R(t – 1, 3)

R(t – 1, 4)

12.9 / PSEUDORANDOM NUMBER GENERATION USING HASH FUNCTIONS 413

Expanding the right-hand side by substituting from Equation 12.1,

D(K, ([A(t) }R(t, n)]⊕ T)) = D(K, ([T⊕ E(K, [A(t - 1) }R(t - 1, 1)])]⊕ T))

Now we recognize that T⊕ T = 0 and that for any x, x⊕ 0 = x. So,

D(K, ([A(t) }R(t, n)]⊕ T)) = D(K, ([E(K, [A(t - 1) }R(t - 1, 1)]))
= A(t - 1) }R(t - 1, 1)

The second part of the demonstration is to show that the for loops in step 2
of the wrap and unwrap algorithms are inverses. For stage k of the wrap algorithm,
the variables R(t - 1, 1) through R(t - 1, n) are input. R(t - 1, 1) is used in the
encryption calculation. R(t - 1, 2) through R(t - 1, n) are mapped, respectively
into R(t, 1) through R(t, n - 1), and R(t, n) is output from the encryption function.
For stage k of the unwrap algorithm, the variables R(t, 1) through R(t, n) are input.
R(t, n) is input to the decryption function to produce R(t - 1, 1). The remaining
variables R(t - 1, 2) through R(t - 1, n) are generated by the for loop, such that
they are mapped, respectively, from R(t, 1) through R(t, n - 1).

Thus, we have shown that the output variables of stage k of the unwrap algo-
rithm equal the input variables of stage k of the wrap algorithm.

12.9 PSEUDORANDOM NUMBER GENERATION USING HASH
FUNCTIONS AND MACs

The essential elements of any pseudorandom number generator (PRNG) are a seed
value and a deterministic algorithm for generating a stream of pseudorandom bits.
If the algorithm is used as a pseudorandom function (PRF) to produce a required
value, such as a session key, then the seed should only be known to the user of the
PRF. If the algorithm is used to produce a stream encryption function, then the seed
has the role of a secret key that must be known to the sender and the receiver.

We noted in Chapters 8 and 10 that, because an encryption algorithm pro-
duces an apparently random output, it can serve as the basis of a (PRNG). Similarly,
a hash function or MAC produces apparently random output and can be used to
build a PRNG. Both ISO standard 18031 (Random Bit Generation) and NIST SP
800-90 (Recommendation for Random Number Generation Using Deterministic
Random Bit Generators) define an approach for random number generation using
a cryptographic hash function. SP 800-90 also defines a random number generator
based on HMAC. We look at these two approaches in turn.

PRNG Based on Hash Function

Figure 12.14a shows the basic strategy for a hash-based PRNG specified in SP 800-
90 and ISO 18031. The algorithm takes as input:

V = seed
seedlen = bit length of V Ú K + 64, where k is a desired security level
expressed in bits

n = desired number of output bits

414 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

The algorithm uses the cryptographic hash function H with an hash value out-
put of outlen bits. The basic operation of the algorithm is

m = <n/outlen=
data = V

W = the null string

For i = 1 to m

wi = H (data)

W = } wi
data = (data + 1) mod 2seedlen

Return leftmost n bits of W

Thus, the pseudorandom bit stream is w1 }w2 } c }wm with the final block
truncated if required.

The SP 800-90 specification also provides for periodically updating V to en-
hance security. The specification also indicates that there are no known or suspected
weaknesses in the hash-based approach for a strong cryptographic hash algorithm,
such as SHA-2.

Figure 12.14 Basic Structure of Hash-Based PRNGs (SP 800-90)

(a) PRNG using cryptographic hash function

(b) PRNG using HMAC

V

K

Cryptographic
hash function

Pseudorandom
output

+1

V

HMAC

Pseudorandom
output

12.9 / PSEUDORANDOM NUMBER GENERATION USING HASH FUNCTIONS 415

PRNG Based on MAC Function

Although there are no known or suspected weaknesses in the use of a cryptographic
hash function for a PRNG in the manner of Figure 12.14a, a higher degree of con-
fidence can be achieved by using a MAC. Almost invariably, HMAC is used for
constructing a MAC-based PRNG. This is because HMAC is a widely used stan-
dardized MAC function and is implemented in many protocols and applications. As
SP 800-90 points out, the disadvantage of this approach compared to the hash-based
approach is that the execution time is twice as long, because HMAC involves two
executions of the underlying hash function for each output block. The advantage of
the HMAC approach is that it provides a greater degree of confidence in its secu-
rity, compared to a pure hash-based approach.

For the MAC-based approach, there are two inputs: a key K and a seed V. In
effect, the combination of K and V form the overall seed for the PRNG specified
in SP 800-90. Figure 12.14b shows the basic structure of the PRNG mechanism, and
the leftmost column of Figure 12.15 shows the logic. Note that the key remains the
same for each block of output, and the data input for each block is equal to the tag
output of the previous block. The SP 800-90 specification also provides for periodi-
cally updating K and V to enhance security.

It is instructive to compare the SP 800-90 recommendation with the use of
HMAC for a PRNG in some applications, and this is shown in Figure 12.15. For the
IEEE 802.11i wireless LAN security standard (Chapter 18), the data input consists
of the seed concatenated with a counter. The counter is incremented for each block
wi of output. This approach would seem to offer enhanced security compared to the
SP 800-90 approach. Consider that for SP 800-90, the data input for output block
wi is just the output wi- 1 of the previous execution of HMAC. Thus, an opponent
who is able to observe the pseudorandom output knows both the input and output
of HMAC. Even so, with the assumption that HMAC is secure, knowledge of the
input and output should not be sufficient to recover K and hence not sufficient to
predict future pseudorandom bits.

The approach taken by the Transport Layer Security protocol (Chapter 17)
and the Wireless Transport Layer Security Protocol (Chapter 18) involves invoking
HMAC twice for each block of output wi. As with IEEE 802.11, this is done in such
a way that the output does not yield direct information about the input. The double
use of HMAC doubles the execution burden and would seem to be security overkill.

Figure 12.15 Three PRNGs Based on HMAC

m = <n/outlen=
w0 = V
W = the null string
For i = 1 to m

wi = MAC(K, wi- 1)
W = W }wi

Return leftmost n bits of W

m = <n/outlen=
W = the null string
For i = 1 to m

wi = MAC(K, (V } i))
W = W }wi

Return leftmost n bits of W

m = <n/outlen=
A(0) = V
W = the null string
For i = 1 to m
A(i) = MAC(K, A(i - 1))
wi = MAC(K, (A(i) }V)
W = W }wi
Return leftmost n bits of W

NIST SP 800-90 IEEE 802.11i TLS/WTLS

Hiva-Network.Com

416 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

12.10 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

authenticator
Cipher-Based Message

Authentication Code
(CMAC)

CMAC
Counter with Cipher Block

Chaining-Message
Authentication Code
(CCM)

cryptographic checksum
cryptographic hash

function
Data Authentication

Algorithm (DAA)
Galois/Counter Mode

(GCM)
HMAC

key encryption key
Key Wrap mode
key wrapping
message authentication
message authentication code

(MAC)

Review Questions

12.1 What types of attacks are addressed by message authentication?
12.2 What two levels of functionality comprise a message authentication or digital signa-

ture mechanism?
12.3 What are some approaches to producing message authentication?
12.4 When a combination of symmetric encryption and an error control code is used for

message authentication, in what order must the two functions be performed?
12.5 What is a message authentication code?
12.6 What is the difference between a message authentication code and a one-way hash

function?
12.7 In what ways can a hash value be secured so as to provide message authentication?
12.8 Is it necessary to recover the secret key in order to attack a MAC algorithm?
12.9 What changes in HMAC are required in order to replace one underlying hash func-

tion with another?

Problems

12.1 If F is an error-detection function, either internal or external use (Figure 12.2) will
provide error-detection capability. If any bit of the transmitted message is altered,
this will be reflected in a mismatch of the received FCS and the calculated FCS,
whether the FCS function is performed inside or outside the encryption function.
Some codes also provide an error-correction capability. Depending on the nature of
the function, if one or a small number of bits is altered in transit, the error-correction
code contains sufficient redundant information to determine the errored bit or bits
and correct them. Clearly, an error-correction code will provide error correction ca-
pability when used external to the encryption function. Will it also provide this capa-
bility if used internal to the encryption function?

12.2 The data authentication algorithm, described in Section 12.6, can be defined as using
the cipher block chaining (CBC) mode of operation of DES with an initialization vec-
tor of zero (Figure 12.7). Show that the same result can be produced using the cipher
feedback mode.

12.10 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 417

12.3 At the beginning of Section 12.6, it was noted that given the CBC MAC of a one-
block message X, say T = MAC(K, X), the adversary immediately knows the CBC
MAC for the two-block message X } (X⊕ T) since this is once again T. Justify this
statement.

12.4 In this problem, we demonstrate that for CMAC, a variant that XORs the second
key after applying the final encryption doesn’t work. Let us consider this for the
case of the message being an integer multiple of the block size. Then, the variant
can be expressed as VMAC(K, M) = CBC(K, M)⊕ K1. Now suppose an adver-
sary is able to ask for the MACs of three messages: the message 0 = 0n, where n is
the cipher block size; the message 1 = 1n; and the message 1 } 0. As a result of these
three queries, the adversary gets T0 = CBC(K, 0)⊕ K1; T1 = CBC(K, 1)⊕ K1 and
T2 = CBC(K, [CBC(K, 1)])⊕ K1. Show that the adversary can compute the correct
MAC for the (unqueried) message 0 } (T0⊕ T1).

12.5 In the discussion of subkey generation in CMAC, it states that the block cipher is ap-
plied to the block that consists entirely of 0 bits. The first subkey is derived from the
resulting string by a left shift of one bit and, conditionally, by XORing a constant that
depends on the block size. The second subkey is derived in the same manner from the
first subkey.
a. What constants are needed for block sizes of 192-bits and 256 bits?
b. Explain how the left shift and XOR accomplishes the desired result.

12.6 Section 12.7 listed four general approaches to provide confidentiality and message
encryption: H S E, A S E, E S A, and E + A.
a. Which of the above performs decryption before verification?
b. Which of the above performs verification before decryption?

12.7 Show that the GHASH function calculates

(X1 # Hm)⊕ (X2 # Hm - 1)⊕ g ⊕ (Xm - 1 # H2)⊕ (Xm # H)

12.8 Draw a figure similar to Figure 12.11 that shows authenticated decryption.
12.9 Alice want to send a single bit of information (a yes or a no) to Bob by means of a

word of length 2. Alice and Bob have four possible keys available to perform mes-
sage authentication. The following matrix shows the 2-bit word sent for each message
under each key:

Message

Key 0 1

1 00 11

2 01 10

3 10 01

4 11 00

a. The preceding matrix is in a useful form for Alice. Construct a matrix with the
same information that would be more useful for Bob.

b. What is the probability that someone else can successfully impersonate Alice?
c. What is the probability that someone can replace an intercepted message with

another message successfully?
12.10 Draw figures similar to Figures 12.12 and 12.13 for the unwrap algorithm.

418 CHAPTER 12 / MESSAGE AUTHENTICATION CODES

12.11 Consider the following key wrapping algorithm:

1. Initialize variables.
A = A6A6A6A6A6A6A6A6
for i = 1 to n
R(i) = Pi

2. Calculate intermediate values.
for j = 0 to 5
for i = 1 to n
B = E(K, [A } R(i)])
t = (n × j) + i
A = t⊕ MSB64(B)
R(i) = LSB64(B)

3. Output results.
C0 = A
for i = 1 to n
Ci = R(i)

a. Compare this algorithm, functionally, with the algorithm specified in SP 800-38F
and described in Section 12.8.

b. Write the corresponding unwrap algorithm.

419

13.1 Digital Signatures

Properties
Attacks and Forgeries
Digital Signature Requirements
Direct Digital Signature

13.2 Elgamal Digital Signature Scheme

13.3 Schnorr Digital Signature Scheme

13.4 NIST Digital Signature Algorithm

The DSA Approach
The Digital Signature Algorithm

13.5 Elliptic Curve Digital Signature Algorithm

Global Domain Parameters
Key Generation
Digital Signature Generation and Authentication

13.6 RSA-PSS Digital Signature Algorithm

Mask Generation Function
The Signing Operation
Signature Verification

13.7 Key Terms, Review Questions, and Problems

CHAPTER

Digital Signatures

420 CHAPTER 13 / DIGITAL SIGNATURES

The most important development from the work on public-key cryptography is the
digital signature. The digital signature provides a set of security capabilities that would
be difficult to implement in any other way.

Figure 13.1 is a generic model of the process of constructing and using digital
signatures. All of the digital signature schemes discussed in this chapter have this
structure. Suppose that Bob wants to send a message to Alice. Although it is not
important that the message be kept secret, he wants Alice to be certain that the
message is indeed from him. For this purpose, Bob uses a secure hash function, such
as SHA-512, to generate a hash value for the message. That hash value, together
with Bob’s private key serves as input to a digital signature generation algorithm,
which produces a short block that functions as a digital signature. Bob sends the
message with the signature attached. When Alice receives the message plus signa-
ture, she (1) calculates a hash value for the message; (2) provides the hash value and
Bob’s public key as inputs to a digital signature verification algorithm. If the algo-
rithm returns the result that the signature is valid, Alice is assured that the message
must have been signed by Bob. No one else has Bob’s private key and therefore no
one else could have created a signature that could be verified for this message with
Bob’s public key. In addition, it is impossible to alter the message without access to
Bob’s private key, so the message is authenticated both in terms of source and in
terms of data integrity.

We begin this chapter with an overview of digital signatures. We then present the
Elgamal and Schnorr digital signature schemes, understanding of which makes it easier
to understand the NIST Digital Signature Algorithm (DSA). The chapter then cov-
ers the two other important standardized digital signature schemes: the Elliptic Curve
Digital Signature Algorithm (ECDSA) and the RSA Probabilistic Signature Scheme
(RSA-PSS).

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

◆ Present an overview of the digital signature process.

◆ Understand the Elgamal digital signature scheme.

◆ Understand the Schnorr digital signature scheme.

◆ Understand the NIST digital signature scheme.

◆ Compare and contrast the NIST digital signature scheme with the Elgamal
and Schnorr digital signature schemes.

◆ Understand the elliptic curve digital signature scheme.

◆ Understand the RSA-PSS digital signature scheme.

13.1 / DIGITAL SIGNATURES 421

13.1 DIGITAL SIGNATURES

Properties

Message authentication protects two parties who exchange messages from any third
party. However, it does not protect the two parties against each other. Several forms
of dispute between the two parties are possible.

Figure 13.1 Simplified Depiction of Essential Elements of Digital Signature Process

Bob Alice

Bob’s
signature

for M

Message M

Cryptographic
hash

function

Digital
signature

generation
algorithm

Digital
signature

verification
algorithm

h

Message M

Cryptographic
hash

function

h

S

Message M S Return
signature

valid or not valid

Bob’s
private

key

(a) Bob signs a message (b) Alice verifies the signature

Bob’s
public
key

422 CHAPTER 13 / DIGITAL SIGNATURES

For example, suppose that John sends an authenticated message to Mary,
using one of the schemes of Figure 12.1. Consider the following disputes that could
arise.

1. Mary may forge a different message and claim that it came from John. Mary
would simply have to create a message and append an authentication code
using the key that John and Mary share.

2. John can deny sending the message. Because it is possible for Mary to forge
a message, there is no way to prove that John did in fact send the message.

Both scenarios are of legitimate concern. Here is an example of the first
scenario: An electronic funds transfer takes place, and the receiver increases the
amount of funds transferred and claims that the larger amount had arrived from
the sender. An example of the second scenario is that an electronic mail message
contains instructions to a stockbroker for a transaction that subsequently turns out
badly. The sender pretends that the message was never sent.

In situations where there is not complete trust between sender and receiver,
something more than authentication is needed. The most attractive solution to
this problem is the digital signature. The digital signature must have the following
properties:

■ It must verify the author and the date and time of the signature.

■ It must authenticate the contents at the time of the signature.

■ It must be verifiable by third parties, to resolve disputes.

Thus, the digital signature function includes the authentication function.

Attacks and Forgeries

[GOLD88] lists the following types of attacks, in order of increasing severity. Here
A denotes the user whose signature method is being attacked, and C denotes the
attacker.

■ Key-only attack: C only knows A’s public key.

■ Known message attack: C is given access to a set of messages and their
signatures.

■ Generic chosen message attack: C chooses a list of messages before attempt-
ing to breaks A’s signature scheme, independent of A’s public key. C then
obtains from A valid signatures for the chosen messages. The attack is generic,
because it does not depend on A’s public key; the same attack is used against
everyone.

■ Directed chosen message attack: Similar to the generic attack, except that the
list of messages to be signed is chosen after C knows A’s public key but before
any signatures are seen.

■ Adaptive chosen message attack: C is allowed to use A as an “oracle.” This
means that C may request from A signatures of messages that depend on
previously obtained message-signature pairs.

13.1 / DIGITAL SIGNATURES 423

[GOLD88] then defines success at breaking a signature scheme as an outcome
in which C can do any of the following with a non-negligible probability:

■ Total break: C determines A’s private key.

■ Universal forgery: C finds an efficient signing algorithm that provides an
equivalent way of constructing signatures on arbitrary messages.

■ Selective forgery: C forges a signature for a particular message chosen by C.

■ Existential forgery: C forges a signature for at least one message. C has
no control over the message. Consequently, this forgery may only be a minor
nuisance to A.

Digital Signature Requirements

On the basis of the properties and attacks just discussed, we can formulate the
following requirements for a digital signature.

■ The signature must be a bit pattern that depends on the message being signed.

■ The signature must use some information only known to the sender to prevent
both forgery and denial.

■ It must be relatively easy to produce the digital signature.

■ It must be relatively easy to recognize and verify the digital signature.

■ It must be computationally infeasible to forge a digital signature, either by
constructing a new message for an existing digital signature or by constructing
a fraudulent digital signature for a given message.

■ It must be practical to retain a copy of the digital signature in storage.

A secure hash function, embedded in a scheme such as that of Figure 13.1, provides
a basis for satisfying these requirements. However, care must be taken in the design
of the details of the scheme.

Direct Digital Signature

The term direct digital signature refers to a digital signature scheme that involves
only the communicating parties (source, destination). It is assumed that the destina-
tion knows the public key of the source.

Confidentiality can be provided by encrypting the entire message plus
signature with a shared secret key (symmetric encryption). Note that it is important
to perform the signature function first and then an outer confidentiality function.
In case of dispute, some third party must view the message and its signature. If the
signature is calculated on an encrypted message, then the third party also needs
a ccess to the decryption key to read the original message. However, if the signature
is the inner operation, then the recipient can store the plaintext message and its
signature for later use in dispute resolution.

The validity of the scheme just described depends on the security of the send-
er’s private key. If a sender later wishes to deny sending a particular message, the
sender can claim that the private key was lost or stolen and that someone else forged
his or her signature. Administrative controls relating to the security of private keys

424 CHAPTER 13 / DIGITAL SIGNATURES

can be employed to thwart or at least weaken this ploy, but the threat is still there,
at least to some degree. One example is to require every signed message to include
a timestamp (date and time) and to require prompt reporting of compromised keys
to a central authority.

Another threat is that a private key might actually be stolen from X at time T.
The opponent can then send a message signed with X’s signature and stamped with
a time before or equal to T.

The universally accepted technique for dealing with these threats is the use
of a digital certificate and certificate authorities. We defer a discussion of this topic
until Chapter 14, and focus in this chapter on digital signature algorithms.

13.2 ELGAMAL DIGITAL SIGNATURE SCHEME

Before examining the NIST Digital Signature Algorithm, it will be helpful to under-
stand the Elgamal and Schnorr signature schemes. Recall from Chapter 10, that the
Elgamal encryption scheme is designed to enable encryption by a user’s public key
with decryption by the user’s private key. The Elgamal signature scheme involves
the use of the private key for digital signature generation and the public key for
digital signature verification [ELGA84, ELGA85].

Before proceeding, we need a result from number theory. Recall from Chapter 2
that for a prime number q, if a is a primitive root of q, then

a, a2, c , aq - 1

are distinct (mod q). It can be shown that, if a is a primitive root of q, then

1. For any integer m, am K 1 (mod q) if and only if m K 0 (mod q - 1).
2. For any integers, i, j, ai K aj (mod q) if and only if i K j (mod q - 1).

As with Elgamal encryption, the global elements of Elgamal digital signature
are a prime number q and a, which is a primitive root of q. User A generates
a private/public key pair as follows.

1. Generate a random integer XA, such that 1 6 XA 6 q - 1.
2. Compute YA = aXA mod q.
3. A’s private key is XA; A’s pubic key is {q, a, YA}.

To sign a message M, user A first computes the hash m = H(M), such that
m is an integer in the range 0 … m … q - 1. A then forms a digital signature as
follows.

1. Choose a random integer K such that 1 … K … q - 1 and gcd(K, q - 1) = 1.
That is, K is relatively prime to q - 1.

2. Compute S1 = aK mod q. Note that this is the same as the computation of C1
for Elgamal encryption.

3. Compute K-1 mod (q - 1). That is, compute the inverse of K modulo q - 1.
4. Compute S2 = K-1(m - XAS1) mod (q - 1).
5. The signature consists of the pair (S1, S2).

Hiva-Network.Com

13.3 / SCHNORR DIGITAL SIGNATURE SCHEME 425

Any user B can verify the signature as follows.

1. Compute V1 = am mod q.
2. Compute V2 = (YA)S1(S1)S2 mod q.

The signature is valid if V1 = V2. Let us demonstrate that this is so. Assume
that the equality is true. Then we have

am mod q = (YA)S1(S1)S2 mod q assume V1 = V2
am mod q = aXAS1aKS2 mod q substituting for YA and S1
am - XAS1 mod q = aKS2 mod q rearranging terms
m - XAS1 K KS2 mod (q - 1) property of primitive roots
m - XAS1 K KK-1 (m - XAS1) mod (q - 1) substituting for S2

For example, let us start with the prime field GF(19); that is, q = 19. It has
primitive roots {2, 3, 10, 13, 14, 15}, as shown in Table 2.7. We choose a = 10.

Alice generates a key pair as follows:

1. Alice chooses XA = 16.
2. Then YA = aXA mod q = a16 mod 19 = 4.
3. Alice’s private key is 16; Alice’s pubic key is {q, a, YA} = {19, 10, 4}.

Suppose Alice wants to sign a message with hash value m = 14.

1. Alice chooses K = 5, which is relatively prime to q - 1 = 18.
2. S1 = aK mod q = 105 mod 19 = 3 (see Table 2.7).
3. K-1 mod (q - 1) = 5-1 mod 18 = 11.
4. S2 = K-1 (m - XAS1) mod (q - 1) = 11 (14 - (16)(3)) mod 18 = -374

mod 18 = 4.

Bob can verify the signature as follows.

1. V1 = am mod q = 1014 mod 19 = 16.
2. V2 = (YA)S1(S1)S2 mod q = (43)(34) mod 19 = 5184 mod 19 = 16.

Thus, the signature is valid because V1 = V2.

13.3 SCHNORR DIGITAL SIGNATURE SCHEME

As with the Elgamal digital signature scheme, the Schnorr signature scheme is
based on discrete logarithms [SCHN89, SCHN91]. The Schnorr scheme minimizes
the message-dependent amount of computation required to generate a signature.
The main work for signature generation does not depend on the message and can
be done during the idle time of the processor. The message-dependent part of the
signature generation requires multiplying a 2n-bit integer with an n-bit integer.

The scheme is based on using a prime modulus p, with p - 1 having a prime
factor q of appropriate size; that is, p - 1 K 0 (mod q). Typically, we use p ≈ 21024
and q ≈ 2160. Thus, p is a 1024-bit number, and q is a 160-bit number, which is also
the length of the SHA-1 hash value.

426 CHAPTER 13 / DIGITAL SIGNATURES

The first part of this scheme is the generation of a private/public key pair,
which consists of the following steps.

1. Choose primes p and q, such that q is a prime factor of p - 1.
2. Choose an integer a, such that aq = 1 mod p. The values a, p, and q comprise a

global public key that can be common to a group of users.

3. Choose a random integer s with 0 6 s 6 q. This is the user’s private key.
4. Calculate v = a-s mod p. This is the user’s public key.

A user with private key s and public key v generates a signature as follows.

1. Choose a random integer r with 0 6 r 6 q and compute x = ar mod p. This
computation is a preprocessing stage independent of the message M to be
signed.

2. Concatenate the message with x and hash the result to compute the value e:

e = H(M }x)

3. Compute y = (r + se) mod q. The signature consists of the pair (e, y).

Any other user can verify the signature as follows.

1. Compute x′ = ayve mod p.
2. Verify that e = H (M }x′).

To see that the verification works, observe that

x′ K ayve K aya-se K ay - se K ar K x (mod p)

Hence, H (M }x′) = H (M }x).

13.4 NIST DIGITAL SIGNATURE ALGORITHM

The National Institute of Standards and Technology (NIST) has published
Federal Information Processing Standard FIPS 186, known as the Digital
Signature Algorithm (DSA). The DSA makes use of the Secure Hash Algorithm
(SHA) described in Chapter 12. The DSA was originally proposed in 1991 and
revised in 1993 in response to public feedback concerning the security of the
scheme. There was a further minor revision in 1996. In 2000, an expanded version
of the standard was issued as FIPS 186-2, subsequently updated to FIPS 186-3 in
2009, and FIPS 186-4 in 2013. This latest version also incorporates digital signa-
ture algorithms based on RSA and on elliptic curve cryptography. In this section,
we discuss DSA.

The DSA Approach

The DSA uses an algorithm that is designed to provide only the digital signa-
ture function. Unlike RSA, it cannot be used for encryption or key exchange.
Nevertheless, it is a public-key technique.

13.4 / NIST DIGITAL SIGNATURE ALGORITHM 427

Figure 13.2 contrasts the DSA approach for generating digital signatures to
that used with RSA. In the RSA approach, the message to be signed is input to a
hash function that produces a secure hash code of fixed length. This hash code is
then encrypted using the sender’s private key to form the signature. Both the mes-
sage and the signature are then transmitted. The recipient takes the message and
produces a hash code. The recipient also decrypts the signature using the sender’s
public key. If the calculated hash code matches the decrypted signature, the signa-
ture is accepted as valid. Because only the sender knows the private key, only the
sender could have produced a valid signature.

The DSA approach also makes use of a hash function. The hash code is pro-
vided as input to a signature function along with a random number k generated for
this particular signature. The signature function also depends on the sender’s private
key (PRa) and a set of parameters known to a group of communicating principals.
We can consider this set to constitute a global public key (PUG).

1 The result is a
signature consisting of two components, labeled s and r.

At the receiving end, the hash code of the incoming message is generated. The
hash code and the signature are inputs to a verification function. The verification
function also depends on the global public key as well as the sender’s public key
(PUa), which is paired with the sender’s private key. The output of the verification
function is a value that is equal to the signature component r if the signature is valid.
The signature function is such that only the sender, with knowledge of the private
key, could have produced the valid signature.

We turn now to the details of the algorithm.

1It is also possible to allow these additional parameters to vary with each user so that they are a part of
a user’s public key. In practice, it is more likely that a global public key will be used that is separate from
each user’s public key.

Figure 13.2 Two Approaches to Digital Signatures

(a) RSA approach

M

H

| | M

Sig Ver

H

Compare

k

s
r

(b) DSA approach

M

H

| | M

E D

H

ComparePRa

PRaPUG PUaPUG

PUa

E(PRa, H(M)]

428 CHAPTER 13 / DIGITAL SIGNATURES

The Digital Signature Algorithm

DSA is based on the difficulty of computing discrete logarithms (see Chapter 2)
and is based on schemes originally presented by Elgamal [ELGA85] and Schnorr
[SCHN91].

Figure 13.3 summarizes the algorithm. There are three parameters that are
public and can be common to a group of users. An N-bit prime number q is chosen.
Next, a prime number p is selected with a length between 512 and 1024 bits such
that q divides (p - 1). Finally, g is chosen to be of the form h(p - 1)/q mod p, where h
is an integer between 1 and (p - 1) with the restriction that g must be greater
than 1.2 Thus, the global public-key components of DSA are the same as in the
Schnorr signature scheme.

With these parameters in hand, each user selects a private key and generates
a public key. The private key x must be a number from 1 to (q - 1) and should
be chosen randomly or pseudorandomly. The public key is calculated from the
private key as y = gx mod p. The calculation of y given x is relatively straight-
forward. However, given the public key y, it is believed to be computationally
infeasible to determine x, which is the discrete logarithm of y to the base g, mod p
(see Chapter 2).

2In number-theoretic terms, g is of order q mod p; see Chapter 2.

Global Public-Key Components

p prime number where 2L - 1 6 p 6 2L
for 512 … L … 1024 and L a multiple of 64;
i.e., bit length L between 512 and 1024 bits
in increments of 64 bits

q prime divisor of (p - 1), where 2N - 1 6 q 6 2N
i.e., bit length of N bits

g = h(p - 1)/q is an exponent mod p,
where h is any integer with 1 6 h 6 (p - 1)
such that h(p - 1)/q mod p 7 1

User’s Private Key

x random or pseudorandom integer with 0 6 x 6 q

User’s Public Key

y = gx mod p

User’s Per-Message Secret Number

k random or pseudorandom integer with 0 6 k 6 q

Signing

r = (gk mod p) mod q

s = [k-1 (H(M) + xr)] mod q

Signature = (r, s)

Verifying

w = (s′)-1 mod q

u1 = [H(M′)w] mod q

u2 = (r′)w mod q

v = [(gu1yu2) mod p] mod q

TEST: v = r′

M = message to be signed

H(M) = hash of M using SHA-1

M′, r′, s′ = received versions of M, r, s

Figure 13.3 The Digital Signature Algorithm (DSA)

13.4 / NIST DIGITAL SIGNATURE ALGORITHM 429

The signature of a message M consists of the pair of numbers r and s, which are
functions of the public key components (p, q, g), the user’s private key (x), the hash
code of the message H(M), and an additional integer k that should be generated
randomly or pseudorandomly and be unique for each signing.

Let M, r′, and s′ be the received versions of M, r, and s, respectively.
Verification is performed using the formulas shown in Figure 13.3. The receiver
generates a quantity v that is a function of the public key components, the sender’s
public key, the hash code of the incoming message, and the received versions of r
and s. If this quantity matches the r component of the signature, then the signature
is validated.

Figure 13.4 depicts the functions of signing and verifying.

Figure 13.4 DSA Signing and Verifying

(a) Signing

(b) Verifying

M

s
r

Mœ

rœ

H

r = (gk mod p) mod q

q

s = [k–1 (H(M) + xr)] mod q

w = (sœ)–1 mod q

k k

k

q

x x

M

H(M)

H
H(Mœ)

p g

q

q

y

v

g

u1 = [H(Mœ)w)] mod q
u2 = (rœ)w mod q
v = [(gu1yu2) mod p] mod q

signature
verification

rœ = v?

rœ

rœ

w

sœ

430 CHAPTER 13 / DIGITAL SIGNATURES

The structure of the algorithm, as revealed in Figure 13.4, is quite interesting.
Note that the test at the end is on the value r, which does not depend on the mes-
sage at all. Instead, r is a function of k and the three global public-key components.
The multiplicative inverse of k (mod q) is passed to a function that also has as inputs
the message hash code and the user’s private key. The structure of this function is
such that the receiver can recover r using the incoming message and signature, the
public key of the user, and the global public key. It is certainly not obvious from
Figure 13.3 or Figure 13.4 that such a scheme would work. A proof is provided in
Appendix K.

Given the difficulty of taking discrete logarithms, it is infeasible for an
opponent to recover k from r or to recover x from s.

Another point worth noting is that the only computationally demanding
task in signature generation is the exponential calculation gk mod p. Because this
value does not depend on the message to be signed, it can be computed ahead of
time. Indeed, a user could precalculate a number of values of r to be used to sign
documents as needed. The only other somewhat demanding task is the determi-
nation of a multiplicative inverse, k-1. Again, a number of these values can be
precalculated.

13.5 ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM

As was mentioned, the 2009 version of FIPS 186 includes a new digital signature
technique based on elliptic curve cryptography, known as the Elliptic Curve Digital
Signature Algorithm (ECDSA). ECDSA is enjoying increasing acceptance due
to the efficiency advantage of elliptic curve cryptography, which yields security
comparable to that of other schemes with a smaller key bit length.

First we give a brief overview of the process involved in ECDSA. In essence,
four elements are involved.

1. All those participating in the digital signature scheme use the same global domain
parameters, which define an elliptic curve and a point of origin on the curve.

2. A signer must first generate a public, private key pair. For the private key, the
signer selects a random or pseudorandom number. Using that random number
and the point of origin, the signer computes another point on the elliptic curve.
This is the signer’s public key.

3. A hash value is generated for the message to be signed. Using the private
key, the domain parameters, and the hash value, a signature is generated. The
signature consists of two integers, r and s.

4. To verify the signature, the verifier uses as input the signer’s public key, the
domain parameters, and the integer s. The output is a value v that is compared
to r. The signature is verified if v = r.

Let us examine each of these four elements in turn.

13.5 / ELLIPTIC CURVE DIGITAL SIGNATURE ALGORITHM 431

Global Domain Parameters

Recall from Chapter 10 that two families of elliptic curves are used in cryptographic
applications: prime curves over Zp and binary curves over GF(2

m). For ECDSA,
prime curves are used. The global domain parameters for ECDSA are the following:

q a prime number

a, b integers that specify the elliptic curve equation defined over Zq with the
equation y2 = x3 + ax + b

G a base point represented by G = (xg, yg) on the elliptic curve equation
n order of point G; that is, n is the smallest positive integer such that

nG = O. This is also the number of points on the curve.

Key Generation

Each signer must generate a pair of keys, one private and one public. The signer,
let us call him Bob, generates the two keys using the following steps:

1. Select a random integer d, d ∈ [1, n - 1]
2. Compute Q = dG. This is a point in Eq(a, b)
3. Bob’s public key is Q and private key is d.

Digital Signature Generation and Authentication

With the public domain parameters and a private key in hand, Bob generates
a digital signature of 320 bytes for message m using the following steps:

1. Select a random or pseudorandom integer k, k ∈ [1, n - 1]
2. Compute point P = (x, y) = kG and r = x mod n. If r = 0 then goto step 1
3. Compute t = k-1 mod n
4. Compute e = H(m), where H is one of the SHA-2 or SHA-3 hash functions.
5. Compute s = k-1(e + dr) mod n. If s = O then goto step 1
6. The signature of message m is the pair (r, s).

Alice knows the public domain parameters and Bob’s public key. Alice is
presented with Bob’s message and digital signature and verifies the signature using
the following steps:

1. Verify that r and s are integers in the range 1 through n - 1
2. Using SHA, compute the 160-bit hash value e = H(m)
3. Compute w = s-1 mod n
4. Compute u1 = ew and u2 = rw
5. Compute the point X = (x1, y1) = u1G + u2Q
6. If X = O, reject the signature else compute v = x1 mod n
7. Accept Bob’s signature if and only if v = r

432 CHAPTER 13 / DIGITAL SIGNATURES

Figure 13.5 illustrates the signature authentication process. We can verify that
this process is valid as follows. If the message received by Alice is in fact signed by
Bob, then

s = k-1(e + dr) mod n

Then

k = s-1(e + dr) mod n
k = (s-1e + s-1dr) mod n
k = (we + wdr) mod n
k = (u1 + u2d) mod n

Now consider that

u1G + u2Q = u1G + u2dG = (u1 + u2d)G = kG

Figure 13.5 ECDSA Signing and Verifying

Q

Yes

No

No

No

No

No

Yes

Yes

Yes

Yes

r, s

Accept
signature

Reject
signature

Bob Alice

Generate k
(x, y) = kG
r = x mod n

Generate private
key d. Public
key Q = dG

q, a, b, G, n
are shared
global variables

Signature of m
is r, s

v = x1 mod n

e = H(m)
s = k–1 (e + dr) mod n

e = H(m)
w = s–1 mod n
u1 = ew, u2 = rw
X = (x1, x2) = u1G + u2Q

r = 0?

s = 0?

X = O?

v = r?

r, s integers
in range
[1, n–1]?

13.6 / RSA-PSS DIGITAL SIGNATURE ALGORITHM 433

In step 6 of the verification process, we have v = x1 mod n, where point
X = (x1, y1) = u1G + u2Q. Thus we see that v = r since r = x mod n and x is the x
coordinate of the point kG and we have already seen that u1G + u2Q = kG.

13.6 RSA-PSS DIGITAL SIGNATURE ALGORITHM

In addition to the NIST Digital Signature Algorithm and ECDSA, the 2009 version
of FIPS 186 also includes several techniques based on RSA, all of which were devel-
oped by RSA Laboratories and are in wide use. A worked-out example, using RSA,
is available at this book’s Web site.

In this section, we discuss the RSA Probabilistic Signature Scheme (RSA-PSS),
which is the latest of the RSA schemes and the one that RSA Laboratories recom-
mends as the most secure of the RSA schemes.

Because the RSA-based schemes are widely deployed in many applications,
including financial applications, there has been great interest in demonstrating that
such schemes are secure. The three main RSA signature schemes differ mainly in
the padding format the signature generation operation employs to embed the hash
value into a message representative, and in how the signature verification opera-
tion determines that the hash value and the message representative are consistent.
For all of the schemes developed prior to PSS, it has not been possible to develop
a mathematical proof that the signature scheme is as secure as the underlying RSA
encryption/decryption primitive [KALI01]. The PSS approach was first proposed by
Bellare and Rogaway [BELL96c, BELL98]. This approach, unlike the other RSA-
based schemes, introduces a randomization process that enables the security of the
method to be shown to be closely related to the security of the RSA algorithm itself.
This makes RSA-PSS more desirable as the choice for RSA-based digital signature
applications.

Mask Generation Function

Before explaining the RSA-PSS operation, we need to describe the mask gener-
ation function (MGF) used as a building block. MGF(X, maskLen) is a pseudo-
random function that has as input parameters a bit string X of any length and the
desired length L in octets of the output. MGFs are typically based on a secure
cryptographic hash function such as SHA-1. An MGF based on a hash function is
intended to be a cryptographically secure way of generating a message digest, or
hash, of variable length based on an underlying cryptographic hash function that
produces a fixed-length output.

The MGF function used in the current specification for RSA-PSS is MGF1,
with the following parameters:

Options Hash hash function with output hLen octets

Input X octet string to be masked

maskLen length in octets of the mask

Output mask an octet string of length maskLen

Hiva-Network.Com

434 CHAPTER 13 / DIGITAL SIGNATURES

MGF1 is defined as follows:

1. Initialize variables.

T = empty string

k = <maskLen/hLen= - 1
2. Calculate intermediate values.

for counter = 0 to k

Represent counter as a 32-bit string C

T = T } Hash(X } C)
3. Output results.

mask = the leading maskLen octets of T

In essence, MGF1 does the following. If the length of the desired output is
equal to the length of the hash value (maskLen = hLen), then the output is the
hash of the input value X concatenated with a 32-bit counter value of 0. If maskLen
is greater than hLen, the MGF1 keeps iterating by hashing X concatenated with the
counter and appending that to the current string T. So that the output is

Hash (X }0) }Hash(X }1) } c }Hash(X }k)

This is repeated until the length of T is greater than or equal to maskLen, at which
point the output is the first maskLen octets of T.

The Signing Operation

MESSAGE ENCODING The first stage in generating an RSA-PSS signature of a message
M is to generate from M a fixed-length message digest, called an encoded message
(EM). Figure 13.6 illustrates this process. We define the following parameters and
functions:

Options Hash hash function with output hLen octets. The current
preferred alternative is SHA-1, which produces a 20-octet
hash value.

MGF mask generation function. The current specification calls
for MGF1.

sLen length in octets of the salt. Typically sLen = hLen, which
for the current version is 20 octets.

Input M message to be encoded for signing.

emBits This value is one less than the length in bits of the RSA
modulus n.

Output EM encoded message. This is the message digest that will be
encrypted to form the digital signature.

Parameters emLen length of EM in octets = <emBits/8= .
padding1 hexadecimal string 00 00 00 00 00 00 00 00; that is, a string

of 64 zero bits.

13.6 / RSA-PSS DIGITAL SIGNATURE ALGORITHM 435

padding2 hexadecimal string of 00 octets with a length
(emLen - sLen - hLen - 2) octets, followed by the
hexadecimal octet with value 01.

salt a pseudorandom number.

bc the hexadecimal value BC.

The encoding process consists of the following steps.

1. Generate the hash value of M: mHash = Hash(M)
2. Generate a pseudorandom octet string salt and form block M′ = padding1 }

mHash } salt
3. Generate the hash value of M′: H = Hash(M′)
4. Form data block DB = padding2 } salt
5. Calculate the MGF value of H: dbMask = MGF(H, emLen - hLen - 1)
6. Calculate maskedDB = DB⊕ dbMsk
7. Set the leftmost 8emLen - emBits bits of the leftmost octet in maskedDB to 0
8. EM = maskedDB }H }0xbc

We make several comments about the complex nature of this message
digest algorithm. All of the RSA-based standardized digital signature schemes
involve appending one or more constants (e.g., padding1 and padding2) in the
process of forming the message digest. The objective is to make it more difficult
for an adversary to find another message that maps to the same message digest

Figure 13.6 RSA-PSS Encoding

Hash

Hash

MGF

M

mHash saltpadding1

bcmaskedDB

saltpadding2

Mœ =

DB =

EM = H

436 CHAPTER 13 / DIGITAL SIGNATURES

as a given message or to find two messages that map to the same message digest.
RSA-PSS also incorporates a pseudorandom number, namely the salt. Because the
salt changes with every use, signing the same message twice using the same private
key will yield two different signatures. This is an added measure of security.

FORMING THE SIGNATURE We now show how the signature is formed by a signer
with private key {d, n} and public key {e, n} (see Figure 9.5). Treat the octet string
EM as an unsigned, nonnegative binary integer m. The signature s is formed by
encrypting m as follows:

s = md mod n

Let k be the length in octets of the RSA modulus n. For example if the key size
for RSA is 2048 bits, then k = 2048/8 = 256. Then convert the signature value s
into the octet string S of length k octets.

Signature Verification

DECRYPTION For signature verification, treat the signature S as an unsigned,
nonnegative binary integer s. The message digest m is recovered by decrypting s as
follows:

m = se mod n

Then, convert the message representative m to an encoded message EM of
length emLen = <(modBits - 1)/8= octets, where modBits is the length in bits of
the RSA modulus n.

EM VERIFICATION EM verification can be described as follows:

Options Hash hash function with output hLen octets.

MGF mask generation function.

sLen length in octets of the salt.

Input M message to be verified.

EM the octet string representing the decrypted signature,
with length emLen = <emBits/8= .

emBits This value is one less than the length in bits of the RSA
modulus n.

Parameters padding1 hexadecimal string 00 00 00 00 00 00 00 00; that is,
a string of 64 zero bits.

padding2 hexadecimal string of 00 octets with a length
(emLen - sLen - hLen - 2) octets, followed by the
hexadecimal octet with value 01.

1. Generate the hash value of M: mHash = Hash(M)
2. If emLen 6 hLen + sLen + 2, output “inconsistent” and stop
3. If the rightmost octet of EM does not have hexadecimal value BC, output

“ inconsistent” and stop

13.6 / RSA-PSS DIGITAL SIGNATURE ALGORITHM 437

4. Let maskedDB be the leftmost emLen - hLen - 1 octets of EM, and let H be
the next hLen octets

5. If the leftmost 8emLen - emBits bits of the leftmost octet in maskedDB are
not all equal to zero, output “inconsistent” and stop

6. Calculate dbMask = MGF (H, emLen - hLen - 1)
7. Calculate DB = maskedDB⊕ dbMsk
8. Set the leftmost 8emLen - emBits bits of the leftmost octet in DB to zero
9. If the leftmost (emLen - hLen - sLen - 1) octets of DB are not equal to

padding2, output “inconsistent” and stop

10. Let salt be the last sLen octets of DB

11. Form block M′ = padding1 }mHash } salt
12. Generate the hash value of M′: H′ = Hash(M′)
13. If H = H′, output “consistent.” Otherwise, output “inconsistent”

Figure 13.7 illustrates the process. The shaded boxes labeled H and H′ cor-
respond, respectively, to the value contained in the decrypted signature and the
value generated from the message M associated with the signature. The remaining
three shaded areas contain values generated from the decrypted signature and com-
pared to known constants. We can now see more clearly the different roles played
by the constants and the pseudorandom value salt, all of which are embedded in the

Figure 13.7 RSA-PSS EM Verification

Hash

Hash

MGF

M

mHash saltpadding1

maskedDB

dbMask

salt

= Mœ

DB =

EM = H

Hœ

438 CHAPTER 13 / DIGITAL SIGNATURES

EM generated by the signer. The constants are known to the verifier, so that the
computed constants can be compared to the known constants as an additional check
that the signature is valid (in addition to comparing H and H′). The salt results in a
different signature every time a given message is signed with the same private key.
The verifier does not know the value of the salt and does not attempt a comparison.
Thus, the salt plays a similar role to the pseudorandom variable k in the NIST DSA
and in ECDSA. In both of those schemes, k is a pseudorandom number generated by
the signer, resulting in different signatures from multiple signings of the same mes-
sage with the same private key. A verifier does not and need not know the value of k.

13.7 KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS

Key Terms

digital signature
Digital Signature Algorithm

(DSA)
direct digital signature

Elgamal digital signature
Elliptic Curve Digital

Signature Algorithm
(ECDSA)

Schnorr digital signature
timestamp

Review Questions
13.1 List two disputes that can arise in the context of message authentication.
13.2 What are the properties a digital signature should have?
13.3 What requirements should a digital signature scheme satisfy?
13.4 What is the difference between direct and arbitrated digital signature?
13.5 In what order should the signature function and the confidentiality function be

applied to a message, and why?
13.6 What are some threats associated with a direct digital signature scheme?

Problems
13.1 Dr. Watson patiently waited until Sherlock Holmes finished. “Some interesting prob-

lem to solve, Holmes?” he asked when Holmes finally logged out.
“Oh, not exactly. I merely checked my email and then made a couple of

network experiments instead of my usual chemical ones. I have only one client now
and I have already solved his problem. If I remember correctly, you once mentioned
cryptology among your other hobbies, so it may interest you.”

“Well, I am only an amateur cryptologist, Holmes. But of course I am interested
in the problem. What is it about?”

“My client is Mr. Hosgrave, director of a small but progressive bank. The bank
is fully computerized and of course uses network communications extensively. The
bank already uses RSA to protect its data and to digitally sign documents that are
communicated. Now the bank wants to introduce some changes in its procedures; in
particular, it needs to digitally sign some documents by two signatories.

1. The first signatory prepares the document, forms its signature, and passes the
document to the second signatory.

13.7 / KEY TERMS, REVIEW QUESTIONS, AND PROBLEMS 439

2. The second signatory as a first step must verify that the document was really signed
by the first signatory. She then incorporates her signature into the document’s sig-
nature so that the recipient, as well as any member of the public, may verify that the
document was indeed signed by both signatories. In addition, only the second signa-
tory has to be able to verify the document’s signature after the first step; that is, the
recipient (or any member of the public) should be able to verify only the complete
document with signatures of both signatories, but not the document in its intermedi-
ate form where only one signatory has signed it. Moreover, the bank would like to
make use of its existing modules that support RSA-style digital signatures.”

“Hm, I understand how RSA can be used to digitally sign documents by one signatory,
Holmes. I guess you have solved the problem of Mr. Hosgrave by appropriate gener-
alization of RSA digital signatures.”

“Exactly, Watson,” nodded Sherlock Holmes. “Originally, the RSA digital sig-
nature was formed by encrypting the document by the signatory’s private decryption
key ‘d’, and the signature could be verified by anyone through its decryption using
publicly known encryption key ‘e’. One can verify that the signature S was formed
by the person who knows d, which is supposed