Undergraduate Texts in Mathematics

Editors

S. AxlerK.A. Ribet

Undergraduate Texts in Mathematics

Abbott: Understanding Analysis.Anglin: Mathematics: A Concise History and

Philosophy.Readings in Mathematics.

Anglin/Lambek: The Heritage of Thales.Readings in Mathematics.

Apostol: Introduction to Analytic Number Theory.Second edition.

Armstrong: Basic Topology.Armstrong: Groups and Symmetry.Axler: Linear Algebra Done Right. Second edition.Beardon: Limits: A New Approach to Real Analysis.Bak/Newman: Complex Analysis. Second edition.Banchoff/Wermer: Linear Algebra Through

Geometry. Second edition.Beck/Robins: Computing the Continuous DiscretelyBerberian: A First Course in Real Analysis.Bix: Conics and Cubics: A Concrete Introduction to

Algebraic Curves. Second edition.Bremaud: An Introduction to Probabilistic Modeling.Bressoud: Factorization and Primality Testing.Bressoud: Second Year Calculus.

Readings in Mathematics.Brickman: Mathematical Introduction to Linear

Programming and Game Theory.Browder: Mathematical Analysis: An Introduction.Buchmann: Introduction to Cryptography. Second

Edition.Buskes/van Rooij: Topological Spaces: From

Distance to Neighborhood.Callahan: The Geometry of Spacetime: An

Introduction to Special and General Relavitity.Carter/van Brunt: The Lebesgue– Stieltjes Integral:

A Practical Introduction.Cederberg: A Course in Modern Geometries. Second

edition.Chambert-Loir: A Field Guide to AlgebraChilds: A Concrete Introduction to Higher Algebra.

Second edition.Chung/AitSahlia: Elementary Probability Theory:

With Stochastic Processes and an Introduction toMathematical Finance. Fourth edition.

Cox/Little/O’Shea: Ideals, Varieties, and Algorithms.Second edition.

Croom: Basic Concepts of Algebraic Topology.Cull/Flahive/Robson: Difference Equations. From

Rabbits to ChaosCurtis: Linear Algebra: An Introductory Approach.

Fourth edition.Daepp/Gorkin: Reading, Writing, and Proving:

A Closer Look at Mathematics.

Devlin: The Joy of Sets: Fundamentalsof-Contemporary Set Theory. Second edition.

Dixmier: General Topology.Driver: Why Math?Ebbinghaus/Flum/Thomas: Mathematical Logic.

Second edition.Edgar: Measure, Topology, and Fractal Geometry.

Second edition.Elaydi: An Introduction to Difference Equations.

Third edition.Erdos/Suranyi: Topics in the Theory of Numbers.Estep: Practical Analysis on One Variable.Exner: An Accompaniment to Higher Mathematics.Exner: Inside Calculus.Fine/Rosenberger: The Fundamental Theory

of Algebra.Fischer: Intermediate Real Analysis.Flanigan/Kazdan: Calculus Two: Linear and

Nonlinear Functions. Second edition.Fleming: Functions of Several Variables. Second

edition.Foulds: Combinatorial Optimization for

Undergraduates.Foulds: Optimization Techniques: An Introduction.Franklin: Methods of Mathematical

Economics.Frazier: An Introduction to Wavelets Through Linear

Algebra.Gamelin: Complex Analysis.Ghorpade/Limaye: A Course in Calculus and Real

AnalysisGordon: Discrete Probability.Hairer/Wanner: Analysis by Its History.

Readings in Mathematics.Halmos: Finite-Dimensional Vector Spaces. Second

edition.Halmos: Naive Set Theory.Hammerlin/Hoffmann: Numerical Mathematics.

Readings in Mathematics.Harris/Hirst/Mossinghoff: Combinatorics and

Graph Theory.Hartshorne: Geometry: Euclid and Beyond.Hijab: Introduction to Calculus and Classical

Analysis. Second edition.Hilton/Holton/Pedersen: Mathematical Reflections:

In a Room with Many Mirrors.Hilton/Holton/Pedersen: Mathematical Vistas: From

a Room with Many Windows.Hoffstein/Pipher/Silverman: An Introduction to

Mathematical Cryptography.Iooss/Joseph: Elementary Stability and Bifurcation

Theory. Second Edition.

(continued after index)

Jeffrey HoffsteinJill PipherJoseph H. Silverman

An Introduction toMathematical Cryptography

123

Jeffrey Hoffstein Jill PipherDepartment of Mathematics Department of MathematicsBrown University Brown University151 Thayer St. 151 Thayer St.Providence, RI 02912 Providence, RI 02912USA USAjhoff@math.brown.edu jpipher@math.brown.edu

Joseph H. SilvermanDepartment of MathematicsBrown University151 Thayer St.Providence, RI 02912USAjhs@math.brown.edu

Editorial BoardS. Axler K.A. RibetMathematics Department Department of MathematicsSan Francisco State University University of CaliforniaSan Francisco, CA 94132 at BerkeleyUSA Berkeley, CA 94720axler@sfsu.edu USA

ribet@math.berkeley.edu

ISBN: 978-0-387-77993-5 e-ISBN: 978-0-387-77994-2DOI: 10.1007/978-0-387-77994-2

Library of Congress Control Number: 2008923038

Mathematics Subject Classification (2000): 94A60, 11T71, 14G50, 68P25

c© 2008 Springer Science+Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec-tion with any form of information storage and retrieval, electronic adaptation, computer software, or bysimilar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.

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Preface

The creation of public key cryptography by Diffie and Hellman in 1976 and thesubsequent invention of the RSA public key cryptosystem by Rivest, Shamir,and Adleman in 1978 are watershed events in the long history of secret com-munications. It is hard to overestimate the importance of public key cryp-tosystems and their associated digital signature schemes in the modern worldof computers and the Internet. This book provides an introduction to thetheory of public key cryptography and to the mathematical ideas underlyingthat theory.

Public key cryptography draws on many areas of mathematics, includingnumber theory, abstract algebra, probability, and information theory. Eachof these topics is introduced and developed in sufficient detail so that thisbook provides a self-contained course for the beginning student. The onlyprerequisite is a first course in linear algebra. On the other hand, studentswith stronger mathematical backgrounds can move directly to cryptographicapplications and still have time for advanced topics such as elliptic curvepairings and lattice-reduction algorithms.

Among the many facets of modern cryptography, this book chooses to con-centrate primarily on public key cryptosystems and digital signature schemes.This allows for an in-depth development of the necessary mathematics re-quired for both the construction of these schemes and an analysis of theirsecurity. The reader who masters the material in this book will not only bewell prepared for further study in cryptography, but will have acquired a realunderstanding of the underlying mathematical principles on which moderncryptography is based.

Topics covered in this book include Diffie–Hellman key exchange, discretelogarithm based cryptosystems, the RSA cryptosystem, primality testing, fac-torization algorithms, probability theory, information theory, collision algo-rithms, elliptic curves, elliptic curve cryptography, pairing-based cryptogra-phy, lattices, lattice-based cryptography, the NTRU cryptosystem, and digi-tal signatures. A final chapter very briefly describes some of the many otheraspects of modern cryptography (hash functions, pseudorandom number gen-erators, zero-knowledge proofs, digital cash, AES,. . . ) and serves to point thereader toward areas for further study.

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vi Preface

Electronic Resources: The interested reader will find additional materialand a list of errata on the Mathematical Cryptography home page:

www.math.brown.edu/~jhs/MathCryptoHome.html

This web page includes many of the numerical exercises in the book, allowingthe reader to cut and paste them into other programs, rather than having toretype them.

No book is ever free from error or incapable of being improved. We wouldbe delighted to receive comments, good or bad, and corrections from ourreaders. You can send mail to us at

mathcrypto@math.brown.edu

Acknowledgments: We, the authors, would like the thank the followingindividuals for test-driving this book and for the many corrections and helpfulsuggestions that they and their students provided: Liat Berdugo, AlexanderCollins, Samuel Dickman, Michael Gartner, Nicholas Howgrave-Graham, Su-Ion Ih, Saeja Kim, Yuji Kosugi, Yesem Kurt, Michelle Manes, Victor Miller,David Singer, William Whyte. In addition, we would like to thank the manystudents at Brown University who took Math 158 and helped us improve theexposition of this book.

Contents

Preface v

Introduction xi

1 An Introduction to Cryptography 11.1 Simple substitution ciphers . . . . . . . . . . . . . . . . . . . . 11.2 Divisibility and greatest common divisors . . . . . . . . . . . . 101.3 Modular arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Prime numbers, unique factorization, and finite fields . . . . . . 261.5 Powers and primitive roots in finite fields . . . . . . . . . . . . 291.6 Cryptography before the computer age . . . . . . . . . . . . . 341.7 Symmetric and asymmetric ciphers . . . . . . . . . . . . . . . . 36Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 Discrete Logarithms and Diffie–Hellman 592.1 The birth of public key cryptography . . . . . . . . . . . . . . 592.2 The discrete logarithm problem . . . . . . . . . . . . . . . . . 622.3 Diffie–Hellman key exchange . . . . . . . . . . . . . . . . . . . 652.4 The ElGamal public key cryptosystem . . . . . . . . . . . . . . 682.5 An overview of the theory of groups . . . . . . . . . . . . . . . 722.6 How hard is the discrete logarithm problem? . . . . . . . . . . 752.7 A collision algorithm for the DLP . . . . . . . . . . . . . . . . 792.8 The Chinese remainder theorem . . . . . . . . . . . . . . . . . 812.9 The Pohlig–Hellman algorithm . . . . . . . . . . . . . . . . . . 862.10 Rings, quotients, polynomials, and finite fields . . . . . . . . . . 92Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3 Integer Factorization and RSA 1133.1 Euler’s formula and roots modulo pq . . . . . . . . . . . . . . . 1133.2 The RSA public key cryptosystem . . . . . . . . . . . . . . . . 1193.3 Implementation and security issues . . . . . . . . . . . . . . . . 1223.4 Primality testing . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.5 Pollard’s p − 1 factorization algorithm . . . . . . . . . . . . . . 133

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viii Contents

3.6 Factorization via difference of squares . . . . . . . . . . . . . . 1373.7 Smooth numbers and sieves . . . . . . . . . . . . . . . . . . . . 1463.8 The index calculus and discrete logarithms . . . . . . . . . . . 1623.9 Quadratic residues and quadratic reciprocity . . . . . . . . . . 1653.10 Probabilistic encryption . . . . . . . . . . . . . . . . . . . . . . 172Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4 Combinatorics, Probability, and Information Theory 1894.1 Basic principles of counting . . . . . . . . . . . . . . . . . . . . 1904.2 The Vigenere cipher . . . . . . . . . . . . . . . . . . . . . . . . 1964.3 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . 2104.4 Collision algorithms and meet-in-the-middle attacks . . . . . . 2274.5 Pollard’s ρ method . . . . . . . . . . . . . . . . . . . . . . . . . 2344.6 Information theory . . . . . . . . . . . . . . . . . . . . . . . . 2434.7 Complexity Theory and P versus NP . . . . . . . . . . . . . . 258Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

5 Elliptic Curves and Cryptography 2795.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 2795.2 Elliptic curves over finite fields . . . . . . . . . . . . . . . . . . 2865.3 The elliptic curve discrete logarithm problem . . . . . . . . . . 2905.4 Elliptic curve cryptography . . . . . . . . . . . . . . . . . . . . 2965.5 The evolution of public key cryptography . . . . . . . . . . . . 3015.6 Lenstra’s elliptic curve factorization algorithm . . . . . . . . . . 3035.7 Elliptic curves over F2 and over F2k . . . . . . . . . . . . . . . 3085.8 Bilinear pairings on elliptic curves . . . . . . . . . . . . . . . . 3155.9 The Weil pairing over fields of prime power order . . . . . . . . 3255.10 Applications of the Weil pairing . . . . . . . . . . . . . . . . . . 334Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

6 Lattices and Cryptography 3496.1 A congruential public key cryptosystem . . . . . . . . . . . . . 3496.2 Subset-sum problems and knapsack cryptosystems . . . . . . . 3526.3 A brief review of vector spaces . . . . . . . . . . . . . . . . . . 3596.4 Lattices: Basic definitions and properties . . . . . . . . . . . . 3636.5 Short vectors in lattices . . . . . . . . . . . . . . . . . . . . . . 3706.6 Babai’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 3796.7 Cryptosystems based on hard lattice problems . . . . . . . . . 3836.8 The GGH public key cryptosystem . . . . . . . . . . . . . . . . 3846.9 Convolution polynomial rings . . . . . . . . . . . . . . . . . . . 3876.10 The NTRU public key cryptosystem . . . . . . . . . . . . . . . 3926.11 NTRU as a lattice cryptosystem . . . . . . . . . . . . . . . . . 4006.12 Lattice reduction algorithms . . . . . . . . . . . . . . . . . . . 4036.13 Applications of LLL to cryptanalysis . . . . . . . . . . . . . . . 418Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

Contents ix

7 Digital Signatures 4377.1 What is a digital signature? . . . . . . . . . . . . . . . . . . . . 4377.2 RSA digital signatures . . . . . . . . . . . . . . . . . . . . . . . 4407.3 ElGamal digital signatures and DSA . . . . . . . . . . . . . . . 4427.4 GGH lattice-based digital signatures . . . . . . . . . . . . . . . 4477.5 NTRU digital signatures . . . . . . . . . . . . . . . . . . . . . . 450Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

8 Additional Topics in Cryptography 4658.1 Hash functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4668.2 Random numbers and pseudorandom number generators . . . . 4688.3 Zero-knowledge proofs . . . . . . . . . . . . . . . . . . . . . . . 4708.4 Secret sharing schemes . . . . . . . . . . . . . . . . . . . . . . . 4738.5 Identification schemes . . . . . . . . . . . . . . . . . . . . . . . 4748.6 Padding schemes and the random oracle model . . . . . . . . . 4768.7 Building protocols from cryptographic primitives . . . . . . . . 4798.8 Hyperelliptic curve cryptography . . . . . . . . . . . . . . . . . 4808.9 Quantum computing . . . . . . . . . . . . . . . . . . . . . . . . 4838.10 Modern symmetric cryptosystems: DES and AES . . . . . . . . 485

List of Notation 489

References 493

Index 501

Introduction

A Principal Goal of (Public Key) Cryptographyis to allow two people to exchange confidential information,even if they have never met and can communicate only viaa channel that is being monitored by an adversary.

The security of communications and commerce in a digital age relies on themodern incarnation of the ancient art of codes and ciphers. Underlying thebirth of modern cryptography is a great deal of fascinating mathematics,some of which has been developed for cryptographic applications, but muchof which is taken from the classical mathematical canon. The principal goalof this book is to introduce the reader to a variety of mathematical topicswhile simultaneously integrating the mathematics into a description of modernpublic key cryptography.

For thousands of years, all codes and ciphers relied on the assumptionthat the people attempting to communicate, call them Bob and Alice, shareda secret key that their adversary, call her Eve, did not possess. Bob woulduse the secret key to encrypt his message, Alice would use the same secretkey to decrypt the message, and poor Eve, not knowing the secret key, wouldbe unable to perform the decryption. A disadvantage of these private keycryptosystems is that Bob and Alice need to exchange the secret key beforethey can get started.

During the 1970s, the astounding idea of public key cryptography burstupon the scene.1 In a public key cryptosystem, Alice has two keys, a publicencryption key KPub and a private (secret) decryption key KPri. Alice pub-lishes her public key KPub, and then Adam and Bob and Carl and everyoneelse can use KPub to encrypt messages and send them to Alice. The ideaunderlying public key cryptgraphy is that although everyone in the worldknows KPub and can use it to encrypt messages, only Alice, who knows theprivate key KPri, is able to decrypt messages.

The advantages of a public key cryptosystem are manifold. For example,Bob can send Alice an encrypted message even if they have never previouslybeen in direct contact. But although public key cryptography is a fascinating

1A brief history of cryptography is given is Sections 1.6, 2.1, 5.5, and 6.7.

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xii Introduction

theoretical concept, it is not at all clear how one might create a public keycryptosystem. It turns out that public key cryptosystems can be based onhard mathematical problems. More precisely, one looks for a mathematicalproblem that is hard to solve a priori, but that becomes easy to solve if oneknows some extra piece of information.

Of course, private key cryptosystems have not disappeared. Indeed, theyare more important than ever, since they tend to be significantly more effi-cient than public key cryptosystems. Thus in practice, if Bob wants to sendAlice a long message, he first uses a public key cryptosystem to send Alicethe key for a private key cryptosystem, and then he uses the private keycryptosystem to encrypt his message. The most efficient modern private keycryptosystems, such as DES and AES, rely for their security on repeated ap-plication of various mixing operations that are hard to unmix without theprivate key. Thus although the subject of private key cryptography is of boththeoretical and practical importance, the connection with fundamental under-lying mathematical ideas is much less pronounced than it is with public keycryptosystems. For that reason, this book concentrates almost exclusively onpublic key cryptography.

Modern mathematical cryptography draws on many areas of mathematics,including especially number theory, abstract algebra (groups, rings, fields),probability, statistics, and information theory, so the prerequisites for studyingthe subject can seem formidable. By way of contrast, the prerequisites forreading this book are minimal, because we take the time to introduce eachrequired mathematical topic in sufficient depth as it is needed. Thus thisbook provides a self-contained treatment of mathematical cryptography forthe reader with limited mathematical background. And for those readers whohave taken a course in, say, number theory or abstract algebra or probability,we suggest briefly reviewing the relevant sections as they are reached and thenmoving on directly to the cryptographic applications.

This book is not meant to be a comprehensive source for all things cryp-tographic. In the first place, as already noted, we concentrate on public keycryptography. But even within this domain, we have chosen to pursue a smallselection of topics to a reasonable mathematical depth, rather than provid-ing a more superficial description of a wider range of subjects. We feel thatany reader who has mastered the material in this book will not only be wellprepared for further study in cryptography, but will have acquired a realunderstanding of the underlying mathematical principles on which moderncryptography is based.

However, this does not mean that the omitted topics are unimportant.It simply means that there is a limit to the amount of material that canbe included in a book (or course) of reasonable length. As in any text, thechoice of particular topics reflects the authors’ tastes and interests. For theconvenience of the reader, the final chapter contains a brief survey of areasfor further study.

Introduction xiii

A Guide to Mathematical Topics: This book includes a significant amountof mathematical material on a variety of topics that are useful in cryptography.The following list is designed to help coordinate the topics that we cover withsubjects that the class or reader may have already studied.

Congruences, primes, and finite fields — §§1.2, 1.3, 1.4, 1.5, 2.10.4The Chinese remainder theorem — §2.8

Euler’s formula — §3.1Primality testing — §3.4

Quadratic reciprocity — §3.9Factorization methods — §§3.5, 3.6, 3.7, 5.6

Discrete logarithms — §§2.2, 3.8, 4.4, 4.5, 5.3Group theory — §2.5

Rings, polynomials, and quotient rings — §2.10, 6.9Combinatorics and probability — §§4.1, 4.3

Information and complexity theory — §§4.6, 4.7Elliptic curves — §§5.1, 5.2, 5.7, 5.8Linear algebra — §6.3

Lattices — §§6.4, 6.5, 6.6, 6.12

Intended Audience and Prerequisites: This book provides a self-con-tained introduction to public key cryptography and to the underlying math-ematics that is required for the subject. It is suitable as a text for advancedundergraduates and beginning graduate students. We provide enough back-ground material so that the book can be used in courses for students with noprevious exposure to abstract algebra or number theory. For classes in whichthe students have a stronger background, the basic mathematical materialmay be omitted, leaving time for some of the more advanced topics.

The formal prerequisites for this book are few, beyond a facility with highschool algebra and, in Chapter 5, analytic geometry. Elementary calculus isused here and there in a minor way, but is not essential, and linear algebrais used in a small way in Chapter 3 and more extensively in Chapter 6. Noprevious knowledge is assumed for mathematical topics such as number the-ory, abstract algebra, and probability theory that play a fundamental role inmodern cryptography. They are covered in detail as needed.

However, it must be emphasized that this is a mathematics book with itsshare of formal definitions and theorems and proofs. Thus it is expected thatthe reader has a certain level of mathematical sophistication. In particular,students who have previously taken a proof-based mathematics course willfind the material easier than those without such background. On the otherhand, the subject of cryptography is so appealing that this book makes agood text for an introduction-to-proofs course, with the understanding thatthe instructor will need to cover the material more slowly to allow the studentstime to become comfortable with proof-based mathematics.

xiv Introduction

Suggested Syllabus: This book contains considerably more material thancan be comfortably covered by beginning students in a one semester course.However, for more advanced students who have already taken courses in num-ber theory and abstract algebra, it should be possible to do most of the remain-ing material. We suggest covering the majority of the topics in Chapters 1, 2,and 3, possibly omitting some of the more technical topics, the optional ma-terial on the Vigenere cipher, and the section on ring theory, which is notused until much later in the book. The next four chapters on information the-ory (Chapter 4), elliptic curves (Chapter 5), lattices (Chapter 6), and digitalsignatures (Chapter 7) are mostly independent of one another, so the instruc-tor has the choice of covering one or two of them in detail or all of them inless depth. We offer the following syllabus as an example of one of the manypossibilities. We have indicated that some sections are optional. Covering theoptional material leaves less time at the end for the later chapters.

Chapter 1 An Introduction to Cryptography.Cover all sections.

Chapter 2 Discrete Logarithms and Diffie–Hellman.Cover Sections 2.1–2.7. Optionally cover the more mathematically so-phisticated Sections 2.8–2.9 on the Pohlig–Hellman algorithm. Omit Sec-tion 2.10 on first reading.

Chapter 3 Integer Factorization and RSA.Cover Sections 3.1–3.5 and Sections 3.9–3.10. Optionally, cover the moremathematically sophisticated Sections 3.6–3.8, dealing with smoothnumbers, sieves, and the index calculus.

Chapter 4 Probability Theory and Information Theory.Cover Sections 4.1, 4.3, and 4.4. Optionally cover the more mathemat-ically sophisticated sections on Pollard’s ρ method (Section 4.5), infor-mation theory (Section 4.6), and complexity theory (Section 4.7). Thematerial on the Vigenere cipher in Section 4.2 nicely illustrates the useof statistics theory in cryptanalysis, but is somewhat off the main path.

Chapter 5 Elliptic Curves.Cover Sections 5.1–5.4. Cover other sections as time permits, but notethat Sections 5.7–5.10 on pairings require finite fields of prime powerorder, which are described in Section 2.10.4.

Chapter 6 Lattices and Cryptography.Cover Sections 6.1–6.8. (If time is short, it is possible to omit eitheror both of Sections 6.1 and 6.2.) Cover either Sections 6.12–6.13 orSections 6.10–6.11, or both, as time permits. Note that Sections 6.10–6.11 on NTRU require the material on polynomial rings and quotientrings covereed in Section 2.10.

Chapter 7 Digital Signatures.Cover Sections 7.1–7.2. Cover the remaining sections as time permits.

Introduction xv

Chapter 8 Additional Topics in Cryptography.The material in this chapter points the reader toward other importantareas of cryptography. It provides a good list of topics and referencesfor student term papers and presentations.

Further Notes for the Instructor: Depending on how much of the hardermathematical material in Chapters 2–4 is covered, there may not be time todelve into both Chapters 5 and 6, so the instructor may need to omit eitherelliptic curves or lattices in order to fit the other material into one semester.

We feel that it is helpful for students to gain an appreciation of the originsof their subject, so we have scattered a handful of sections throughout the bookcontaining some brief comments on the history of cryptography. Instructorswho want to spend more time on mathematics may omit these sections withoutaffecting the mathematical narrative.