## Cryptology and Computational Number TheoryIn the past dozen or so years, cryptology and computational number theory have become increasingly intertwined. Because the primary cryptologic application of number theory is the apparent intractability of certain computations, these two fields could part in the future and again go their separate ways. But for now, their union is continuing to bring ferment and rapid change in both subjects. This book contains the proceedings of an AMS Short Course in Cryptology and Computational Number Theory, held in August 1989 during the Joint Mathematics Meetings in Boulder, Colorado. These eight papers by six of the top experts in the field will provide readers with a thorough introduction to some of the principal advances in cryptology and computational number theory over the past fifteen years. In addition to an extensive introductory article, the book contains articles on primality testing, discrete logarithms, integer factoring, knapsack cryptosystems, pseudorandom number generators, the theoretical underpinnings of cryptology, and other number theory-based cryptosystems. Requiring only background in elementary number theory, this book is aimed at nonexperts, including graduate students and advanced undergraduates in mathematics and computer science. |

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### Contents

Cryptology and Computational Number TheoryAn Introduction | 1 |

Primality Testing | 13 |

Factoring | 27 |

The Discrete Logarithm Problem | 49 |

The Rise and Fall of Knapsack Cryptosystems | 75 |

The Search for Provably Secure Cryptosystems | 89 |

Pseudorandom Number Generators in Cryptography and Number Theory | 115 |

Odds and Ends from Cryptology and Computational Number Theory | 145 |

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### Common terms and phrases

Adleman adversary applications attack basic Blum Carl Pomerance choose cipher text coin tosses Comp complexity composite computational number theory Computer Science computing discrete logarithms congruence congruential cryptographic cryptology decrypt defined Diffie-Hellman digits discrete logarithm problem divisor easy efficient elliptic curve encryption scheme exponentiation finite fields Foundations of Computer GF(p given Goldreich Goldwasser Hellman heuristic input invert iteration Jacobi sum knapsack cryptosystems knapsack problem Lagarias Lecture Notes Lenstra linear linear congruential Math mathematics Merkle-Hellman Micali modulo multiplication N-PRG nonuniform Notes in Computer Odlyzko one-way function output pair polynomial Pomerance predict primality testing prime factor Probabilistic encryption probability Proc proof system protocol proved pseudorandom bit pseudorandom number public key quadratic residue random number running secret key security parameter sequence Shamir sieve solution solve square theorem Theory of Computing trapdoor trapdoor functions Turing machine values vector Z/nZ zero-knowledge