2nd Edition

# Elliptic Curves Number Theory and Cryptography, Second Edition

**Also available as eBook on:**

Like its bestselling predecessor, **Elliptic Curves: Number Theory and Cryptography, Second Edition** develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves.

**New to the Second Edition**

Taking a basic approach to elliptic curves, this accessible book prepares readers to tackle more advanced problems in the field. It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality testing. The book also discusses the use of elliptic curves in Fermat’s Last Theorem. Relevant abstract algebra material on group theory and fields can be found in the appendices.

**INTRODUCTION**

**THE BASIC THEORY**

Weierstrass Equations

The Group Law

Projective Space and the Point at Infinity

Proof of Associativity

Other Equations for Elliptic Curves

Other Coordinate Systems

The j-Invariant

Elliptic Curves in Characteristic 2

Endomorphisms

Singular Curves

Elliptic Curves mod n

**TORSION POINTS**

Torsion Points

Division Polynomials

The Weil Pairing

The Tate–Lichtenbaum Pairing

**Elliptic Curves over Finite Fields**

Examples

The Frobenius Endomorphism

Determining the Group Order

A Family of Curves

Schoof’s Algorithm

Supersingular Curves

**The Discrete Logarithm Problem**

The Index Calculus

General Attacks on Discrete Logs

Attacks with Pairings

Anomalous Curves

Other Attacks

**Elliptic Curve Cryptography**

The Basic Setup

Diffie–Hellman Key Exchange

Massey–Omura Encryption

ElGamal Public Key Encryption

ElGamal Digital Signatures

The Digital Signature Algorithm

ECIES

A Public Key Scheme Based on Factoring

A Cryptosystem Based on the Weil Pairing

**Other Applications**

Factoring Using Elliptic Curves

Primality Testing

**Elliptic Curves over Q**

The Torsion Subgroup: The Lutz–Nagell Theorem

Descent and the Weak Mordell–Weil Theorem

Heights and the Mordell–Weil Theorem

Examples

The Height Pairing

Fermat’s Infinite Descent

2-Selmer Groups; Shafarevich–Tate Groups

A Nontrivial Shafarevich–Tate Group

Galois Cohomology

**Elliptic Curves over C**

Doubly Periodic Functions

Tori Are Elliptic Curves

Elliptic Curves over C

Computing Periods

Division Polynomials

The Torsion Subgroup: Doud’s Method

**Complex Multiplication**

Elliptic Curves over C

Elliptic Curves over Finite Fields

Integrality of j-Invariants

Numerical Examples

Kronecker’s Jugendtraum

**DIVISORS**

Definitions and Examples

The Weil Pairing

The Tate–Lichtenbaum Pairing

Computation of the Pairings

Genus One Curves and Elliptic Curves

Equivalence of the Definitions of the Pairings

Nondegeneracy of the Tate–Lichtenbaum Pairing

**ISOGENIES**

The Complex Theory

The Algebraic Theory

Vélu’s Formulas

Point Counting

Complements

**Hyperelliptic Curves**

Basic Definitions

Divisors

Cantor’s Algorithm

The Discrete Logarithm Problem

**Zeta Functions**

Elliptic Curves over Finite Fields

Elliptic Curves over Q

**Fermat’s Last Theorem**

Overview

Galois Representations

Sketch of Ribet’s Proof

Sketch of Wiles’s Proof

**APPENDIX A: NUMBER THEORY**

**APPENDIX B: GROUPS**

**APPENDIX C: FIELDS**

**APPENDIX D: COMPUTER**

**packages**

**REFERENCES**

**INDEX**

*Exercises appear at the end of each chapter.*

### Biography

Lawrence C. Washington

… the book is well structured and does not waste the reader’s time in dividing cryptography from number theory-only information. This enables the reader just to pick the desired information. … a very comprehensive guide on the theory of elliptic curves. … I can recommend this book for both cryptographers and mathematicians doing either their Ph.D. or Master’s … I enjoyed reading and studying this book and will be glad to have it as a future reference.

—IACR book reviews, April 2010

Praise for the First EditionThere are already a number of books about elliptic curves, but this new offering by Washington is definitely among the best of them. It gives a rigorous though relatively elementary development of the theory of elliptic curves, with emphasis on those aspects of the theory most relevant for an understanding of elliptic curve cryptography. … an excellent companion to the books of Silverman and Blake, Seroussi and Smart. It would be a fine asset to any library or collection.

—Mathematical Reviews, Issue 2004e

Washington … has found just the right level of abstraction for a first book … . Notably, he offers the most lucid and concrete account ever of the perpetually mysterious Shafarevich–Tate group. A pleasure to read! Summing Up: Highly recommended.

—CHOICE, March 2004

… a nice, relatively complete, elementary account of elliptic curves.

—Bulletin of the AMS