Computational/Algorithmic Number Theory

Abstract

Computational and algorithmic number theory are two very closely related subjects; they are both concerned with, among many others, computer algorithms, particularly efficient algorithms (including parallel and distributed algorithms, sometimes also including computer architectures), for solving different sorts of problems in number theory and in other areas, including computing and cryptography. Primality testing, integer factorization and discrete logarithms are, amongst many others, the most interesting, difficult and useful problems in number theory, computing and cryptography. In this chapter, we shall study both computational and algorithmic aspects of number theory. More specifically, we shall study various algorithms for primality testing, integer factorization and discrete logarithms that are particularly applicable and useful in computing and cryptography, as well as methods for many other problems in number theory, such as the Goldbach conjecture and the odd perfect number problem.