Elliptic Curves Number Theory and Cryptography

ISBN-10: 1420071467
ISBN-13: 9781420071467
Edition: 2nd 2008 (Revised)
List price: $115.95 Buy it from $108.10
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Description: Like its bestselling predecessor, this book develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. This edition now includes alternative coordinate systems and related computational  More...

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Book details

List price: $115.95
Edition: 2nd
Copyright year: 2008
Publisher: CRC Press LLC
Publication date: 4/3/2008
Binding: Hardcover
Pages: 536
Size: 6.25" wide x 9.75" long x 1.00" tall
Weight: 2.134
Language: English

Like its bestselling predecessor, this book develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. This edition now includes alternative coordinate systems and related computational issues, a more elementary treatment of the TateLichtenbaum pairing, additional elliptic curve cryptosystems, and Douds analytic method for computing torsion on elliptic curves over Q. It also contains new chapters on isogenies and hyperelliptic curves as well as discusses how to compute elliptic curves in some popular computer algebra systems. Basic exercises appear at the end of each chapter, with selected answers in an appendix.

Introduction
Exercises
The Basic Theory
Weierstrass Equations
The Group Law
Projective Space and the Point at Infinity
Proof of Associativity
The Theorems of Pappus and Pascal
Other Equations for Elliptic Curves
Legendre Equation
Cubic Equations
Quartic Equations
Intersection of Two Quadratic Surfaces
Other Coordinate Systems
Projective Coordinates
Jacobian Coordinates
Edwards Coordinates
The j-invariant
Elliptic Curves in Characteristic 2
Endomorphisms
Singular Curves
Elliptic Curves mod n
Exercises
Torsion Points
Torsion Points
Division Polynomials
The Weil Pairing
The Tate-Lichtenbaum Pairing
Exercises
Elliptic Curves over Finite Fields
Examples
The Frobenius Endomorphism
Determining the Group Order
Subfield Curves
Legendre Symbols
Orders of Points
Baby Step, Giant Step
A Family of Curves
Schoof's Algorithm
Supersingular Curves
Exercises
The Discrete Logarithm Problem
The Index Calculus
General Attacks on Discrete Logs
Baby Step, Giant Step
Pollard's [rho] and [lambda] Methods
The Pohlig-Hellman Method
Attacks with Pairings
The MOV Attack
The Frey-Ruck Attack
Anomalous Curves
Other Attacks
Exercises
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
Exercises
Other Applications
Factoring Using Elliptic Curves
Primality Testing
Exercises
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
Exercises
Elliptic Curves over C
Doubly Periodic Functions
Tori are Elliptic Curves
Elliptic Curves over C
Computing Periods
The Arithmetic-Geometric Mean
Division Polynomials
The Torsion Subgroup: Doud's Method
Exercises
Complex Multiplication
Elliptic Curves over C
Elliptic Curves over Finite Fields
Integrality of j-invariants
Numerical Examples
Kronecker's Jugendtraum
Exercises
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
The Weil Pairing
The Tate-Lichtenbaum Pairing
Nondegeneracy of the Tate-Lichtenbaum Pairing
Exercises
Isogenies
The Complex Theory
The Algebraic Theory
Velu's Formulas
Point Counting
Complements
Exercises
Hyperelliptic Curves
Basic Definitions
Divisors
Cantor's Algorithm
The Discrete Logarithm Problem
Exercises
Zeta Functions
Elliptic Curves over Finite Fields
Elliptic Curves over Q
Exercises
Fermat's Last Theorem
Overview
Galois Representations
Sketch of Ribet's Proof
Sketch of Wiles's Proof
Number Theory
Groups
Fields
Computer Packages
Pari
Magma
SAGE
References
Index

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