Nµ T

®¤ C°§°

S ®¤ E ¤© ©®

© 2008 by Taylor & Francis Group, LLC

DISCRETE

MATHEMATICS

ITS APPLICATIONS

Series Editor

Kenneth H. Rosen, Ph.D.

Juergen Bierbrauer, Introduction to Coding Theory

Francine Blanchet-Sadri, Algorithmic Combinatorics on Partial Words

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, Handbook of Combinatorial Designs, Second Edition

Martin Erickson and Anthony Vazzana, Introduction to Number Theory

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, Combinatorial Methods with Computer Applications

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

© 2008 by Taylor & Francis Group, LLC

Continued Titles

William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization

Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration

and Search

Charles C. Lindner and Christopher A. Rodgers, Design Theory

Hang T. Lau, A Java Library of Graph Algorithms and Optimization

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, Second Edition

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

W. D. Wallis, Introduction to Combinatorial Designs, Second Edition

Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition

© 2008 by Taylor & Francis Group, LLC

DISCRETE MATHEMATICS AND ITS APPLICATIONS

Series Editor KENNETH H. ROSEN

E¬¬©°© Cµ

Nµ T

®¤ C°§°

S ®¤ E ¤© ©®

L AW RENCE C . WA S HI NG TON

Unive rsit y of M ary l and

College Par k, M aryl and, U.S.A.

© 2008 by Taylor & Francis Group, LLC

Chapman & Hall/CRC

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2008 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-13: 978-1-4200-7146-7 (Hardcover)

This book contains information obtained from authentic and highly regarded sources Reason-

able efforts have been made to publish reliable data and information, but the author and publisher

cannot assume responsibility for the validity of all materials or the consequences of their use. The

Authors and Publishers have attempted to trace the copyright holders of all material reproduced

in this publication and apologize to copyright holders if permission to publish in this form has not

been obtained. If any copyright material has not been acknowledged please write and let us know so

we may rectify in any future reprint

Except as permitted under U.S. Copyright Law, 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

Washington, Lawrence C.

Elliptic curves : number theory and cryptography / Lawrence C. Washington.

-- 2nd ed.

p. cm. -- (Discrete mathematics and its applications ; 50)

Includes bibliographical references and index.

ISBN 978-1-4200-7146-7 (hardback : alk. paper)

1. Curves, Elliptic. 2. Number theory. 3. Cryptography. I. Title. II. Series.

QA567.2.E44W37 2008

516.3™52--dc22 2008006296

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

© 2008 by Taylor & Francis Group, LLC

To Susan and Patrick

© 2008 by Taylor & Francis Group, LLC

Preface

Over the last two or three decades, elliptic curves have been playing an in-

creasingly important role both in number theory and in related ¬elds such as

cryptography. For example, in the 1980s, elliptic curves started being used

in cryptography and elliptic curve techniques were developed for factorization

and primality testing. In the 1980s and 1990s, elliptic curves played an impor-

tant role in the proof of Fermat™s Last Theorem. The goal of the present book

is to develop the theory of elliptic curves assuming only modest backgrounds

in elementary number theory and in groups and ¬elds, approximately what

would be covered in a strong undergraduate or beginning graduate abstract

algebra course. In particular, we do not assume the reader has seen any al-

gebraic geometry. Except for a few isolated sections, which can be omitted

if desired, we do not assume the reader knows Galois theory. We implicitly

use Galois theory for ¬nite ¬elds, but in this case everything can be done

explicitly in terms of the Frobenius map so the general theory is not needed.

The relevant facts are explained in an appendix.

The book provides an introduction to both the cryptographic side and the

number theoretic side of elliptic curves. For this reason, we treat elliptic curves

over ¬nite ¬elds early in the book, namely in Chapter 4. This immediately

leads into the discrete logarithm problem and cryptography in Chapters 5, 6,

and 7. The reader only interested in cryptography can subsequently skip to

Chapters 11 and 13, where the Weil and Tate-Lichtenbaum pairings and hy-

perelliptic curves are discussed. But surely anyone who becomes an expert in

cryptographic applications will have a little curiosity as to how elliptic curves

are used in number theory. Similarly, a non-applications oriented reader could

skip Chapters 5, 6, and 7 and jump straight into the number theory in Chap-

ters 8 and beyond. But the cryptographic applications are interesting and

provide examples for how the theory can be used.

There are several ¬ne books on elliptic curves already in the literature. This

book in no way is intended to replace Silverman™s excellent two volumes [109],

[111], which are the standard references for the number theoretic aspects of

elliptic curves. Instead, the present book covers some of the same material,

plus applications to cryptography, from a more elementary viewpoint. It is

hoped that readers of this book will subsequently ¬nd Silverman™s books more

accessible and will appreciate their slightly more advanced approach. The

books by Knapp [61] and Koblitz [64] should be consulted for an approach to

the arithmetic of elliptic curves that is more analytic than either this book or

[109]. For the cryptographic aspects of elliptic curves, there is the recent book

of Blake et al. [12], which gives more details on several algorithms than the

ix

© 2008 by Taylor & Francis Group, LLC

x

present book, but contains few proofs. It should be consulted by serious stu-

dents of elliptic curve cryptography. We hope that the present book provides

a good introduction to and explanation of the mathematics used in that book.

The books by Enge [38], Koblitz [66], [65], and Menezes [82] also treat elliptic

curves from a cryptographic viewpoint and can be pro¬tably consulted.

Notation. The symbols Z, Fq , Q, R, C denote the integers, the ¬nite

¬eld with q elements, the rationals, the reals, and the complex numbers,

respectively. We have used Zn (rather than Z/nZ) to denote the integers

mod n. However, when p is a prime and we are working with Zp as a ¬eld,

rather than as a group or ring, we use Fp in order to remain consistent with

the notation Fq . Note that Zp does not denote the p-adic integers. This

choice was made for typographic reasons since the integers mod p are used

frequently, while a symbol for the p-adic integers is used only in a few examples

in Chapter 13 (where we use Op ). The p-adic rationals are denoted by Qp .

If K is a ¬eld, then K denotes an algebraic closure of K. If R is a ring, then

R— denotes the invertible elements of R. When K is a ¬eld, K — is therefore

the multiplicative group of nonzero elements of K. Throughout the book,

the letters K and E are generally used to denote a ¬eld and an elliptic curve

(except in Chapter 9, where K is used a few times for an elliptic integral).

Acknowledgments. The author thanks Bob Stern of CRC Press for

suggesting that this book be written and for his encouragement, and the

editorial sta¬ at CRC Press for their help during the preparation of the book.

Ed Eikenberg, Jim Owings, Susan Schmoyer, Brian Conrad, and Sam Wagsta¬

made many suggestions that greatly improved the manuscript. Of course,

there is always room for more improvement. Please send suggestions and

corrections to the author (lcw@math.umd.edu). Corrections will be listed on

the web site for the book (www.math.umd.edu/∼lcw/ellipticcurves.html).

© 2008 by Taylor & Francis Group, LLC

Preface to the Second Edition

The main question asked by the reader of a preface to a second edition is

“What is new?” The main additions are the following:

1. A chapter on isogenies.

2. A chapter on hyperelliptic curves, which are becoming prominent in

many situations, especially in cryptography.

3. A discussion of alternative coordinate systems (projective coordinates,

Jacobian coordinates, Edwards coordinates) and related computational

issues.

4. A more complete treatment of the Weil and Tate-Lichtenbaum pairings,

including an elementary de¬nition of the Tate-Lichtenbaum pairing, a

proof of its nondegeneracy, and a proof of the equality of two common

de¬nitions of the Weil pairing.

5. Doud™s analytic method for computing torsion on elliptic curves over Q.

6. Some additional techniques for determining the group of points for an

elliptic curve over a ¬nite ¬eld.

7. A discussion of how to do computations with elliptic curves in some

popular computer algebra systems.

8. Several more exercises.

Thanks are due to many people, especially Susan Schmoyer, Juliana Belding,

Tsz Wo Nicholas Sze, Enver Ozdemir, Qiao Zhang,and Koichiro Harada for

helpful suggestions. Several people sent comments and corrections for the ¬rst

edition, and we are very thankful for their input. We have incorporated most

of these into the present edition. Of course, we welcome comments and correc-

tions for the present edition (lcw@math.umd.edu). Corrections will be listed

on the web site for the book (www.math.umd.edu/∼lcw/ellipticcurves.html).

xi

© 2008 by Taylor & Francis Group, LLC

Suggestions to the Reader

This book is intended for at least two audiences. One is computer scientists

and cryptographers who want to learn about elliptic curves. The other is for

mathematicians who want to learn about the number theory and geometry of

elliptic curves. Of course, there is some overlap between the two groups. The

author of course hopes the reader wants to read the whole book. However, for

those who want to start with only some of the chapters, we make the following

suggestions.

Everyone: A basic introduction to the subject is contained in Chapters 1,

2, 3, 4. Everyone should read these.

I. Cryptographic Track: Continue with Chapters 5, 6, 7. Then go to

Chapters 11 and 13.

II. Number Theory Track: Read Chapters 8, 9, 10, 11, 12, 14, 15. Then

go back and read the chapters you skipped since you should know how the

subject is being used in applications.

III. Complex Track: Read Chapters 9 and 10, plus Section 12.1.

xiii

© 2008 by Taylor & Francis Group, LLC

Contents

1 Introduction 1

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Basic Theory 9

2.1 Weierstrass Equations . . . . . . . . . . . . . . . . . . . . . . 9

2.2 The Group Law . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Projective Space and the Point at In¬nity . . . . . . . . . . . 18

2.4 Proof of Associativity . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 The Theorems of Pappus and Pascal . . . . . . . . . . 33

2.5 Other Equations for Elliptic Curves . . . . . . . . . . . . . . 35

2.5.1 Legendre Equation . . . . . . . . . . . . . . . . . . . . 35

2.5.2 Cubic Equations . . . . . . . . . . . . . . . . . . . . . 36

2.5.3 Quartic Equations . . . . . . . . . . . . . . . . . . . . 37

2.5.4 Intersection of Two Quadratic Surfaces . . . . . . . . 39

2.6 Other Coordinate Systems . . . . . . . . . . . . . . . . . . . 42

2.6.1 Projective Coordinates . . . . . . . . . . . . . . . . . . 42

2.6.2 Jacobian Coordinates . . . . . . . . . . . . . . . . . . 43

2.6.3 Edwards Coordinates . . . . . . . . . . . . . . . . . . 44

2.7 The j-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.8 Elliptic Curves in Characteristic 2 . . . . . . . . . . . . . . . 47

2.9 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.10 Singular Curves . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.11 Elliptic Curves mod n . . . . . . . . . . . . . . . . . . . . . . 64

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Torsion Points 77

3.1 Torsion Points . . . . ..... . . . . . . . . . . . . . . . . . 77

3.2 Division Polynomials ..... . . . . . . . . . . . . . . . . . 80

3.3 The Weil Pairing . . . ..... . . . . . . . . . . . . . . . . . 86

3.4 The Tate-Lichtenbaum Pairing . . . . . . . . . . . . . . . . . 90

Exercises . . . . . . . . . . ..... . . . . . . . . . . . . . . . . . 92

4 Elliptic Curves over Finite Fields 95

4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 The Frobenius Endomorphism . . . . . . . . . . . . . . . . . 98

4.3 Determining the Group Order . . . . . . . . . . . . . . . . . 102

4.3.1 Sub¬eld Curves . . . . . . . . . . . . . . . . . . . . . . 102

xv

© 2008 by Taylor & Francis Group, LLC

xvi

4.3.2 Legendre Symbols . . . . . . . . . . . . . . . . . . . . 104

4.3.3 Orders of Points . . . . . . . . . . . . . . . . . . . . . 106

4.3.4 Baby Step, Giant Step . . . . . . . . . . . . . . . . . . 112

4.4 A Family of Curves . . . . . . . . . . . . . . . . . . . . . . . 115

4.5 Schoof™s Algorithm . . . . . . . . . . . . . . . . . . . . . . . 123

4.6 Supersingular Curves . . . . . . . . . . . . . . . . . . . . . . 130

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5 The Discrete Logarithm Problem 143

5.1 The Index Calculus . . . . . . . . . . . . . . . . . . . . . . . 144

5.2 General Attacks on Discrete Logs . . . . . . . . . . . . . . . 146

5.2.1 Baby Step, Giant Step . . . . . . . . . . . . . . . . . . 146

5.2.2 Pollard™s ρ and » Methods . . . . . . . . . . . . . . . . 147

5.2.3 The Pohlig-Hellman Method . . . . . . . . . . . . . . 151

5.3 Attacks with Pairings . . . . . . . . . . . . . . . . . . . . . . 154

5.3.1 The MOV Attack . . . . . . . . . . . . . . . . . . . . . 154

5.3.2 The Frey-R¨ck Attack . . . .

u . . . . . . . . . . . . . . 157

5.4 Anomalous Curves . . . . . . . . . . . . . . . . . . . . . . . . 159

5.5 Other Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6 Elliptic Curve Cryptography 169

6.1 The Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.2 Di¬e-Hellman Key Exchange . . . . . . . . . . . . . . . . . . 170

6.3 Massey-Omura Encryption . . . . . . . . . . . . . . . . . . . 173

6.4 ElGamal Public Key Encryption ..... . . . . . . . . . . 174

6.5 ElGamal Digital Signatures . . . . . . . . . . . . . . . . . . . 175

6.6 The Digital Signature Algorithm . . . . . . . . . . . . . . . . 179

6.7 ECIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.8 A Public Key Scheme Based on Factoring . . . . . . . . . . . 181

6.9 A Cryptosystem Based on the Weil Pairing . . . . . . . . . . 184

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7 Other Applications 189

7.1 Factoring Using Elliptic Curves . . . . . . . . . . . . . . . . 189

7.2 Primality Testing . . . . . . . . . . . . . . . . . . . . . . . . 194

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

8 Elliptic Curves over Q 199

8.1 The Torsion Subgroup. The Lutz-Nagell Theorem . . . . . . 199

8.2 Descent and the Weak Mordell-Weil Theorem . . . . . . . . 208

8.3 Heights and the Mordell-Weil Theorem . . . . . . . . . . . . 215

8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8.5 The Height Pairing . . . . . . . . . . . . . . . . . . . . . . . 230

8.6 Fermat™s In¬nite Descent . . . . . . . . . . . . . . . . . . . . 231

© 2008 by Taylor & Francis Group, LLC

xvii

8.7 2-Selmer Groups; Shafarevich-Tate Groups . . . . . . . . . . 236

8.8 A Nontrivial Shafarevich-Tate Group . . . . . . . . . . . . . 239

8.9 Galois Cohomology . . . . . . . . . . . . . . . . . . . . . . . 244

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

9 Elliptic Curves over C 257

9.1 Doubly Periodic Functions . . . . . . . . . . . . . . . . . . . 257

9.2 Tori are Elliptic Curves . . . . . . . . . . . . . . . . . . . . . 267

9.3 Elliptic Curves over C . . . . . . . . . . . . . . . . . . . . . . 272

9.4 Computing Periods . . . . . . . . . . . . . . . . . . . . . . . 286

9.4.1 The Arithmetic-Geometric Mean . . . . . . . . . . . . 288

9.5 Division Polynomials . . . . . . . . . . . . . . . . . . . . . . 294

9.6 The Torsion Subgroup: Doud™s Method . . . . . . . . . . . . 302

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

10 Complex Multiplication 311

10.1 Elliptic Curves over C . . .... . . . . . . . . . . . . . . . . 311

10.2 Elliptic Curves over Finite Fields . . . . . . . . . . . . . . . . 318

10.3 Integrality of j-invariants .... . . . . . . . . . . . . . . . . 322

10.4 Numerical Examples . . . .... . . . . . . . . . . . . . . . . 330

10.5 Kronecker™s Jugendtraum .... . . . . . . . . . . . . . . . . 336

Exercises . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . 337

11 Divisors 339

11.1 De¬nitions and Examples . . . . . . . . . . . . . . . . . . . . 339

11.2 The Weil Pairing . . . . . . . . . . . . . . . . . . . . . . . . . 349

11.3 The Tate-Lichtenbaum Pairing . . . . . . . . . . . . . . . . . 354

11.4 Computation of the Pairings . . . . . . . . . . . . . . . . . . 358

11.5 Genus One Curves and Elliptic Curves . . . . . . . . . . . . 364

11.6 Equivalence of the De¬nitions of the Pairings . . . . . . . . . 370

11.6.1 The Weil Pairing . . . . . . . . . . . . . . . . . . . . . 371

11.6.2 The Tate-Lichtenbaum Pairing . . . . . . . . . . . . . 374

11.7 Nondegeneracy of the Tate-Lichtenbaum Pairing . . . . . . . 375

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

12 Isogenies 381

12.1 The Complex Theory . . . . . . . . . . . . . . . . . . . . . . 381

12.2 The Algebraic Theory . . . . . . . . . . . . . . . . . . . . . . 386

12.3 V´lu™s Formulas . . .

e . . . . . . . . . . . . . . . . . . . . . . 392

12.4 Point Counting . . . . . . . . . . . . . . . . . . . . . . . . . . 396

12.5 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

© 2008 by Taylor & Francis Group, LLC

xviii

13 Hyperelliptic Curves 407

13.1 Basic De¬nitions . . . . . . . . . . . . . . . . . . . . . . . . . 407

13.2 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

13.3 Cantor™s Algorithm . . . . . . . . . . . . . . . . . . . . . . . 417

13.4 The Discrete Logarithm Problem . . . . . . . . . . . . . . . . 420

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

14 Zeta Functions 429

14.1 Elliptic Curves over Finite Fields . . . . . . . . . . . . . . . . 429

14.2 Elliptic Curves over Q . . . . . . . . . . . . . . . . . . . . . . 433

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

15 Fermat™s Last Theorem 445

15.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

15.2 Galois Representations . . . . . . . . . . . . . . . . . . . . . 448

15.3 Sketch of Ribet™s Proof . . . . . . . . . . . . . . . . . . . . . 454

15.4 Sketch of Wiles™s Proof . . . . . . . . . . . . . . . . . . . . . 461

A Number Theory 471

B Groups 477

C Fields 481

D Computer Packages 489

D.1 Pari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

D.2 Magma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

D.3 SAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

References 501

© 2008 by Taylor & Francis Group, LLC