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

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

Chapters on isogenies and hyperelliptic curves
A discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational issues
A more complete treatment of the Weil and Tate–Lichtenbaum pairings
Doud’s analytic method for computing torsion on elliptic curves over Q
An explanation of how to perform calculations with elliptic curves in several popular computer algebra systems
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.