Investigation and implementation of some probabilistic algorithms.

Abstract

Probabilistic algorithms were shown to be of importance after some fast polynomial time probabilistic solutions to some "hard" problems were given. In this thesis, theoretical foundations of probabilistic algorithms are studied. Probabilistic solutions to some "hard" problems are studied and investigated. Also probabilistic algorithms for some "easy problems" are studied and presented. The main concentration is on the probabilistic solutions to primality testing. The probability of a composite integer passing the Fermat test is derived. This includes an upper bound on the probability of a given integer being a Carmichael number. Three applications of this work are given as follows: 1. Using the pseduoprimlity test alone to test for primality. 2. Using the pseduoprimlity test with the Rabin test to test for Carmichael numbers. 3. Using the pseduoprimality test to generate keys for the Pohlig-Hellman encryption scheme.