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