## Elliptic Curves: Number Theory and CryptographyElliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students. Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired. By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory. |

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### Contents

Introduction | v |

Exercises | 8 |

The Basic Theory | 9 |

22 The Group Law | 12 |

23 Projective Space and the Point at Infinity | 18 |

24 Proof of Associativity | 20 |

241 The Theorems of Pappus and Pascal | 32 |

25 Other Equations for Elliptic Curves | 35 |

Exercises | 175 |

Other Applications | 179 |

72 Primality Testing | 184 |

Exercises | 187 |

Elliptic Curves over Q | 189 |

82 Descent and the Weak MordellWeil Theorem | 198 |

83 Heights and the MordellWeil Theorem | 206 |

84 Examples | 214 |

253 Quartic Equations | 36 |

254 Intersection of Two Quadratic Surfaces | 39 |

26 The jinvariant | 41 |

27 Elliptic Curves in Characteristic 2 | 44 |

28 Endomorphisms | 46 |

29 Singular Curves | 55 |

210 Elliptic Curves mod n | 59 |

Exercises | 67 |

Torsion Points | 73 |

32 Division Polynomials | 76 |

33 The Weil Pairing | 82 |

Exercises | 86 |

Elliptic Curves over Finite Fields | 89 |

42 The Frobenius Endomorphism | 92 |

43 Determining the Group Order | 96 |

432 Legendre Symbols | 98 |

433 Orders of Points | 100 |

434 Baby Step Giant Step | 103 |

44 A Family of Curves | 105 |

45 Schoofs Algorithm | 113 |

46 Supersingular Curves | 120 |

Exercises | 130 |

The Discrete Logarithm Problem | 133 |

51 The Index Calculus | 134 |

52 General Attacks on Discrete Logs | 136 |

522 Pollards p and A Methods | 137 |

523 The PohligHellman Method | 141 |

53 The MOV Attack | 144 |

54 Anomalous Curves | 147 |

55 The TateLichtenbaum Pairing | 153 |

56 Other Attacks | 156 |

Elliptic Curve Cryptography | 159 |

62 DiffieHellman Key Exchange | 160 |

63 MasseyOmura Encryption | 163 |

64 ElGamal Public Key Encryption | 164 |

65 ElGamal Digital Signatures | 165 |

66 The Digital Signature Algorithm | 168 |

67 A Public Key Scheme Based on Factoring | 169 |

68 A Cryptosystem Based on the Weil Pairing | 173 |

85 The Height Pairing | 221 |

86 Fermats Infinite Descent | 222 |

87 2Selmer Groups ShafarevichTate Groups | 227 |

88 A Nontrivial ShafarevichTate Group | 229 |

89 Galois Cohomology | 234 |

Exercises | 244 |

Elliptic Curves over C | 247 |

92 Tori are Elliptic Curves | 257 |

93 Elliptic Curves over C | 262 |

94 Computing Periods | 275 |

941 The ArithmeticGeometric Mean | 277 |

95 Division Polynomials | 283 |

Exercises | 291 |

Complex Multiplication | 295 |

102 Elliptic Curves over Finite Fields | 302 |

103 Integrality of jinvariants | 306 |

104 Numerical Examples | 314 |

105 Kroneckers Jugendtraum | 320 |

Exercises | 321 |

Divisors | 323 |

112 The Weil Pairing | 333 |

113 The TateLichtenbaum Pairing | 338 |

114 Computation of the Pairings | 341 |

115 Genus One Curves and Elliptic Curves | 346 |

Exercises | 353 |

Zeta Functions | 355 |

122 Elliptic Curves over Q | 359 |

Exercises | 368 |

Fermats Last Theorem | 371 |

132 Galois Representations | 374 |

133 Sketch of Ribets Proof | 380 |

134 Sketch of Wiless Proof | 387 |

Number Theory | 397 |

Groups | 403 |

Fields | 407 |

415 | |

425 | |

### Other editions - View all

Elliptic Curves: Number Theory and Cryptography, Second Edition Lawrence C. Washington Limited preview - 2008 |

Elliptic Curves: Number Theory and Cryptography, Second Edition Lawrence C. Washington No preview available - 2008 |

### Common terms and phrases

algebraic closure algebraic integer algorithm Alice assume automorphism calculation characteristic choose coefficients common root complex multiplication compute coordinates Corollary corresponds cubic curve over Q defined over Q denominator denote discrete log problem dividing divisor doubly periodic E(Fp E(Fq elements elliptic curve defined elliptic curve y2 endomorphism ring example Exercise exists Fermat's finite field follows formulas Frobenius Galois given by y2 group law Hasse's theorem hence homomorphism implies intersection isomorphic j-invariant lattice Lemma method mod q modular form Mordell-Weil theorem nonzero number theory obtain p-adic p-adic integers point of order points at infinity pole polynomial positive integer PROOF Let Proposition proved rational functions rational number reduction root of unity satisfies says Section Show Similarly square mod subgroup supersingular Suppose surjective tangent line Tate-Lichtenbaum pairing torsion points torsion subgroup Weierstrass equation Weierstrass form Weil pairing x-coordinate yields zero