Elliptic Curves
Number Theory and Cryptography, Second Edition

Table of Contents

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.