Elliptic curves : number theory and cryptography /
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| Author / Creator: | Washington, Lawrence C. |
|---|---|
| Edition: | 2nd ed. |
| Imprint: | Boca Raton, FL : Chapman & Hall/CRC, c2008. |
| Description: | xviii, 513 p. : ill. ; 25 cm. |
| Language: | English |
| Series: | Discrete mathematics and its applications |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/7136306 |
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 [rho] and [lambda] Methods
- 5.2.3. The Pohlig-Hellman Method
- 5.3. Attacks with Pairings
- 5.3.1. The MOV Attack
- 5.3.2. The Frey-Ruck 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. Velu'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
