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Elliptic Curves-- Number Theory and Cryptography(Second Edition).pdf

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    Preface
    Preface to the Second Edition
    Contents
    Chapter 1 Introduction
    Exercises
    Chapter 2 The Basic Theory
    2.1 Weierstrass Equations
    2.2 The Group Law
    2.3 Projective Space and the Point at In.nity
    2.4 Proof of Associativity
    2.5 Other Equations for Elliptic Curves
    2.6 Other Coordinate Systems
    2.7 The j-invariant
    2.8 Elliptic Curves in Characteristic 2
    2.9 Endomorphisms
    2.10 Singular Curves
    2.11 Elliptic Curves mod n
    Exercises
    Chapter 3 Torsion Points
    3.1 Torsion Points
    3.2 Division Polynomials
    3.3 The Weil Pairing
    3.4 The Tate-Lichtenbaum Pairing
    Exercises
    Chapter 4 Elliptic Curves over Finite Fields
    4.1 Examples
    4.2 The Frobenius Endomorphism
    4.3 Determining the Group Order
    4.4 A Family of Curves
    4.5 Schoof ’s Algorithm
    4.6 Supersingular Curves
    Exercises
    Chapter 5 The Discrete Logarithm Problem
    5.1 The Index Calculus
    5.2 General Attacks on Discrete Logs
    5.3 Attacks with Pairings
    5.4 Anomalous Curves
    5.5 Other Attacks
    Exercises
    Chapter 6 Elliptic Curve Cryptography
    6.1 The Basic Setup
    6.2 Di.e-Hellman Key Exchange
    6.3 Massey-Omura Encryption
    6.4 ElGamal Public Key Encryption
    6.5 ElGamal Digital Signatures
    6.6 The Digital Signature Algorithm
    6.7 ECIES
    6.8 A Public Key Scheme Based on Factoring
    6.9 A Cryptosystem Based on the Weil Pairing
    Exercises
    Chapter 7 Other Applications
    7.1 Factoring Using Elliptic Curves
    7.2 Primality Testing
    Exercises
    Chapter 8 Elliptic Curves over Q
    8.1 The Torsion Subgroup. The Lutz-Nagell Theorem
    8.2 Descent and the Weak Mordell-Weil Theorem
    8.3 Heights and the Mordell-Weil Theorem
    8.4 Examples
    8.5 The Height Pairing
    8.6 Fermat’s In.nite Descent
    8.7 2-Selmer Groups; Shafarevich-Tate Groups
    8.8 A Nontrivial Shafarevich-Tate Group
    8.9 Galois Cohomology
    Exercises
    Chapter 9 Elliptic Curves over C
    9.1 Doubly Periodic Functions
    9.2 Tori are Elliptic Curves
    9.3 Elliptic Curves over C
    9.4 Computing Periods
    9.5 Division Polynomials
    9.6 The Torsion Subgroup: Doud’s Method
    Exercises
    Chapter 10 Complex Multiplication
    10.1 Elliptic Curves over C
    10.2 Elliptic Curves over Finite Fields
    10.3 Integrality of j-invariants
    10.4 Numerical Examples
    10.5 Kronecker’s Jugendtraum
    Exercises
    Chapter 11 Divisors
    11.1 De.nitions and Examples
    11.2 The Weil Pairing
    11.3 The Tate-Lichtenbaum Pairing
    11.4 Computation of the Pairings
    11.5 Genus One Curves and Elliptic Curves
    11.6 Equivalence of the De.nitions of the Pairings
    11.7 Nondegeneracy of the Tate-Lichtenbaum Pairing
    Exercises
    Chapter 12 Isogenies
    12.1 The Complex Theory
    12.2 The Algebraic Theory
    12.3 V´elu’s Formulas
    12.4 Point Counting
    12.5 Complements
    Exercises
    Chapter 13 Hyperelliptic Curves
    13.1 Basic De.nitions
    13.2 Divisors
    13.3 Cantor’s Algorithm
    13.4 The Discrete Logarithm Problem
    Exercises
    Chapter 14 Zeta Functions
    14.1 Elliptic Curves over Finite Fields
    14.2 Elliptic Curves over Q
    Exercises
    Chapter 15 Fermat’s Last Theorem
    15.1 Overview
    15.2 Galois Representations
    15.3 Sketch of Ribet’s Proof
    15.4 Sketch of Wiles’s Proof
    Appendix A Number Theory
    Basic results
    p-adic numbers
    Appendix B Groups
    Basic de.nitions
    Structure theorems
    Homomorphisms
    Appendix C Fields
    Finite .elds
    Embeddings and automorphisms
    Appendix D Computer Packages
    D.1 Pari
    D.2 Magma
    D.3 SAGE
    Elliptic Curves Number Theory and Cryptography Second Edition © 2008 by Taylor & Francis Group, LLC
    DISCRETE MATHEMATICS ITS APPLICATIONS © 2008 by Taylor & Francis Group, LLC
    Continued Titles © 2008 by Taylor & Francis Group, LLC
    DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN Elliptic Curves Number Theory and Cryptography Second Edition LAWRENCE C. WASHINGTON University of Maryland College Park, Maryland, 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. -- 2nd ed. Elliptic curves : number theory and cryptography / Lawrence C. Washington. 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. 2008006296 QA567.2.E44W37 2008 516.3’52--dc22 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 fields 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 fields, 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 finite fields, 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 finite fields 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 fine 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 find 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 © 2008 by Taylor & Francis Group, LLC ix
    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 profitably consulted. Notation. The symbols Z, Fq, Q, R, C denote the integers, the finite field 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 field, 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 field, then K denotes an algebraic closure of K. If R is a ring, then × denotes the invertible elements of R. When K is a field, K × is therefore R the multiplicative group of nonzero elements of K. Throughout the book, the letters K and E are generally used to denote a field 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 staff at CRC Press for their help during the preparation of the book. Ed Eikenberg, Jim Owings, Susan Schmoyer, Brian Conrad, and Sam Wagstaff 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
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