An Introduction to Cryptography, Second Edition

394 Pages·2007·3.62 MB·English

DISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor KENNETH H. ROSEN
An INTRODUCTION to
CRYPTOGRAPHY
Second Edition
© 2007 by Taylor & Francis Group, LLC
DISCRETE
MATHEMATICS
and
ITS APPLICATIONS
Series Editor
Kenneth H. Rosen, Ph.D.
Juergen Bierbrauer, Introduction to Coding Theory
Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems
Charalambos A. Charalambides, Enumerative Combinatorics
Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography
Charles J. Colbourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs
Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses,
Constructions, and Existence
Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders
Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry,
Second Edition
Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition
Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory
Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information
Theory and Data Compression, Second Edition
Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability:
Experiments with a Symbolic Algebra Environment
Leslie Hogben, Handbook of Linear Algebra
Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory
David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and
Nonorientable Surfaces
Richard E. Klima, Neil P. Sigmon, and Ernest L. Stitzinger, Applications of Abstract Algebra
with Maple™ and MATLAB®, Second Edition
Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science
and Engineering
William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization
Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration
and Search
© 2007 by Taylor & Francis Group, LLC
Continued Titles
Charles C. Lindner and Christopher A. Rodgers, Design Theory
Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied
Cryptography
Richard A. Mollin, Algebraic Number Theory
Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times
Richard A. Mollin, Fundamental Number Theory with Applications
Richard A. Mollin, An Introduction to Cryptography, Second Edition
Richard A. Mollin, Quadratics
Richard A. Mollin, RSA and Public-Key Cryptography
Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers
Dingyi Pei, Authentication Codes and Combinatorial Designs
Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics
Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary
Approach
Jörn Steuding, Diophantine Analysis
Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition
Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and
Coding Design
Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography
© 2007 by Taylor & Francis Group, LLC
DISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor KENNETH H. ROSEN
An INTRODUCTION to
CRYPTOGRAPHY
Second Edition
RICHARD A. MOLLIN
Boca Raton London New York
Chapman & Hall/CRC is an imprint of the
Taylor & Francis Group, an informa business
© 2007 by Taylor & Francis Group, LLC
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2007 by Taylor & Francis Group, LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number-10: 1-58488-618-8 (Hardcover)
International Standard Book Number-13: 978-1-58488-618-1 (Hardcover)
This book contains information obtained from authentic and highly regarded sources. Reprinted
material is quoted with permission, and sources are indicated. A wide variety of references are
listed. Reasonable efforts have been made to publish reliable data and information, but the author
and the publisher cannot assume responsibility for the validity of all materials or for the conse-
quences of their use.
No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any
electronic, mechanical, or other means, now known or hereafter invented, including photocopying,
microfilming, and recording, or in any information storage or retrieval system, without written
permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.
copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC)
222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that
provides licenses and registration for a variety of users. For organizations that have been granted a
photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and
are used only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Mollin, Richard A., 1947-
An Introduction to Cryptography / Richard A. Mollin. -- 2nd ed.
p. cm. -- (Discrete mathematics and its applications)
Includes bibliographical references and index.
ISBN-13: 978-1-58488-618-1 (acid-free paper)
ISBN-10: 1-58488-618-8 (acid-free paper)
1. Coding theory--Textbooks. I. Title. II. Series.
QA268.M65 2007
003’.54--dc22 2006049639
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
© 2007 by Taylor & Francis Group, LLC
To Kathleen Ellen — my Soul Mate.
© 2007 by Taylor & Francis Group, LLC
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Mathematical Basics 1
1.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Primes, Primality Testing, and Induction . . . . . . . . . . 6
1.3 An Introduction to Congruences . . . . . . . . . . . . . . . . 17
1.4 Euler, Fermat, and Wilson . . . . . . . . . . . . . . . . . . . 35
1.5 Primitive Roots . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.6 The Index Calculus and Power Residues . . . . . . . . . . 51
1.7 Legendre, Jacobi, & Quadratic Reciprocity . . . . . . . . . 58
1.8 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2 Cryptographic Basics 79
2.1 Definitions and Illustrations . . . . . . . . . . . . . . . . . . 79
2.2 Classic Ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.3 Stream Ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.4 LFSRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.5 Modes of Operation . . . . . . . . . . . . . . . . . . . . . . . . 122
2.6 Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3 DES and AES 131
3.1 S-DES and DES . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.2 AES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4 Public-Key Cryptography 157
4.1 The Ideas Behind PKC . . . . . . . . . . . . . . . . . . . . . 157
4.2 Digital Envelopes and PKCs . . . . . . . . . . . . . . . . . . 165
4.3 RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.4 ElGamal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.5 DSA — The DSS . . . . . . . . . . . . . . . . . . . . . . . . . 187
5 Primality Testing 189
5.1 True Primality Tests . . . . . . . . . . . . . . . . . . . . . . . 189
5.2 Probabilistic Primality Tests . . . . . . . . . . . . . . . . . . 198
vii
© 2007 by Taylor & Francis Group, LLC
viii
5.3 Recognizing Primes . . . . . . . . . . . . . . . . . . . . . . . . 204
6 Factoring 207
6.1 Classical Factorization Methods . . . . . . . . . . . . . . . . 207
6.2 The Continued Fraction Algorithm . . . . . . . . . . . . . . 211
6.3 Pollard’s Algorithms . . . . . . . . . . . . . . . . . . . . . . . 214
6.4 The Quadratic Sieve . . . . . . . . . . . . . . . . . . . . . . . 217
6.5 The Elliptic Curve Method (ECM) . . . . . . . . . . . . . . 220
7 Electronic Mail and Internet Security 223
7.1 History of the Internet and the WWW . . . . . . . . . . . 223
7.2 Pretty Good Privacy (PGP) . . . . . . . . . . . . . . . . . . 227
7.3 Protocol Layers and SSL . . . . . . . . . . . . . . . . . . . . . 241
7.4 Internetworking and Security — Firewalls . . . . . . . . . 250
7.5 Client–Server Model and Cookies . . . . . . . . . . . . . . . 259
8 Leading-Edge Applications 263
8.1 Login and Network Security . . . . . . . . . . . . . . . . . . 263
8.2 Viruses and Other Infections . . . . . . . . . . . . . . . . . . 273
8.3 Smart Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
8.4 Biometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
Appendix A: Fundamental Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Appendix B: Computer Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Appendix C: The Rijndael S-Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
Appendix D: Knapsack Ciphers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
Appendix E: Silver-Pohlig-Hellman Algorithm . . . . . . . . . . . . . . 344
Appendix F: SHA-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
Appendix G: Radix-64 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Appendix H: Quantum Cryptography. . . . . . . . . . . . . . . . . . . . . . . . 352
Solutions to Odd-Numbered Exercises . . . . . . . . . . . . . . . . . . . . . . . 358
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
© 2007 by Taylor & Francis Group, LLC
Preface
The second edition of the original introductory undergraduate text for a
one-semester course in cryptography is redesigned to be more accessible. This
includes the decision to include many items of contemporary interest not con-
tained in the first edition, such as electronic mail and Internet security, and some
leading-edge applications. The former comprises the history of the WWW, PGP,
protocol layers, SSL, firewalls, client-server models, and cookies, all contained
in Chapter 7. The latter encompasses login and network security, viruses and
other computer infections, as well as smart cards and biometrics, making up
the closing Chapter 8 of the main text. In the appendices, we retained the data
on fundamental mathematical facts. However, instead of leading each chapter
with mathematical background to each of the cryptographic concepts, we have
placed all mathematical basics in Chapter 1, and we have placed all crypto-
graphic basics in Chapter 2. In this fashion, all essential background material
is grounded at the outset.
Symmetric and public-key cryptosystems comprise Chapters 3 and 4, re-
spectively, with the addition of the digital signature standard at the end of
Chapter 4, not contained in the first edition. In order to make the presenta-
tion of DES more palatable to the reader, we have included a new discussion of
S-DES (“baby DES”) as a preamble to DES in Chapter 3.
We maintain the coverage of factoring and primality testing in Chapters 5
and 6, respectively. However, we include a wealth of new aspects of “recogniz-
ing” primes in Chapter 5, including the recent discovery of an unconditional
deterministic polynomial-time algorithm for primality testing. Furthermore,
instead of the more advanced number field sieve, which we have excluded in
this edition, we have placed the elliptic curve method in Chapter 6. We have,
nevertheless, excluded the chapter on advanced topics — the more advanced
elliptic curve cryptography, the coverage of zero knowledge — and have placed
quantum cryptography in an appendix but deleted the more advanced expo-
sition on quantum computing. This has reduced the number of entries in the
bibliography because the first edition had a large number of references to those
advanced topics and points to the greater accessibility of this edition. We have
added Pollard’s two algorithms, the p−1 and rho factoring methods in Chapter
6, and lead the chapter with classical factoring methods with more breadth than
the first edition.
Other than Appendix A on mathematical facts, we have included eight other
appendices on computer arithmetic, which was part of Chapter 1 of the first edi-
tion; the Rijndael S-Box, also an appendix in the first edition; knapsack ciphers,
which was part of Chapter 3 of the first edition; the Silver-Pohlig-Hellman Algo-
rithm; the SHA-1 algorithm; and radix-64 encoding, the latter three not included
in the first edition, and quantum cryptography in the concluding Appendix H.
The numbering system has been changed from the global approach in the
first edition to the standard numbering found in most texts. The use of footnotes
has been curtailed in this edition. For instance, the mini-biographies are placed
ix
© 2007 by Taylor & Francis Group, LLC