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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
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Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration
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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
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Jörn Steuding, Diophantine Analysis
Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition
Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and
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DISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor KENNETH H. ROSEN




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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
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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


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© 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


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© 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