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
- 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.
Author(s): Lawrence C. Washington (Author)
Edition: 2
Publisher: Chapman and Hall/CRC
Year: 2008
Language: English
City: New York
Cover
Title Page
Copyright Page
Dedication Page
Preface
Preface to Second Edition
Suggestions to Reader
Table of Contents
1 Introduction
Exercises
2 The Basic Theory
2.1 Weierstrass Equations
2.2 The Group Law
2.3 Projective Space and the Point at Infinity
2.4 Proof of Associativity
2.4.1 The Theorems of Pappus and Pascal
2.5 Other Equations for Elliptic Curves
2.5.1 Legendre Equation
2.5.2 Cubic Equations
2.5.3 Quartic Equations
2.5.4 Intersection of Two Quadratic Surfaces
2.6 Other Coordinate Systems
2.6.1 Projective Coordinates
2.6.2 Jacobian Coordinates
2.6.3 Edwards Coordinates
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
3 Torsion Points
3.1 Torsion Points
3.2 Division Polynomials
3.3 The Weil Pairing
3.4 The Tate-Lichtenbaum Pairing
Exercises
4 Elliptic Curves over Finite Fields
4.1 Examples
4.2 The Frobenius Endomorphism
4.3 Determining the Group Order
4.3.1 Subfield Curves
4.3.2 Legendre Symbols
4.3.3 Orders of Points
4.3.4 Baby Step, Giant Step
4.4 A Family of Curves
4.5 Schoof’s Algorithm
4.6 Supersingular Curves
Exercises
5 The Discrete Logarithm Problem
5.1 The Index Calculus
5.2 General Attacks on Discrete Logs
5.2.1 Baby Step, Giant Step
5.2.2 Pollard’s ρ and λ Methods
5.2.3 The Pohlig-Hellman Method
5.3 Attacks with Pairings
5.3.1 The MOV Attack
5.3.2 The Frey-Rück Attack
5.4 Anomalous Curves
5.5 Other Attacks
Exercises
6 Elliptic Curve Cryptography
6.1 The Basic Setup
6.2 Diffie-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
7 Other Applications
7.1 Factoring Using Elliptic Curves
7.2 Primality Testing
Exercises
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 Infinite Descent
8.7 2-Selmer Groups; Shafarevich-Tate Groups
8.8 A Nontrivial Shafarevich-Tate Group
8.9 Galois Cohomology
Exercises
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
9.4.1 The Arithmetic-Geometric Mean
9.5 Division Polynomials
9.6 The Torsion Subgroup: Doud’s Method
Exercises
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
11 Divisors
11.1 Definitions 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 Definitions of the Pairings
11.6.1 The Weil Pairing
11.6.2 The Tate-Lichtenbaum Pairing
11.7 Nondegeneracy of the Tate-Lichtenbaum Pairing
Exercises
12 Isogenies
12.1 The Complex Theory
12.2 The Algebraic Theory
12.3 Vélu’s Formulas
12.4 Point Counting
12.5 Complements
Exercises
13 Hyperelliptic Curves
13.1 Basic Definitions
13.2 Divisors
13.3 Cantor’s Algorithm
13.4 The Discrete Logarithm Problem
Exercises
14 Zeta Functions
14.1 Elliptic Curves over Finite Fields
14.2 Elliptic Curves over Q
Exercises
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
A Number Theory
B Groups
C Fields
D Computer Packages
D.1 Pari
D.2 Magma
D.3 SAGE
References
Index
