CHAPTER 12 - Algorithms in Number Theory

Abstract

This chapter discusses algorithms that solve two basic problems in computational number theory—factoring integers into prime factors and finding discrete logarithms. In the factoring problem, one is given an integer n1 and is asked to find the decomposition of n into prime factors. It is common to split this problem into two parts. The first is called primality testing: given n, it is determined whether n is prime or composite. The second is called factorization: if n is composite, a nontrivial divisor of n is to be calculated. In the discrete logarithm problem, one is given a prime number p, and two elements h, y of the multiplicative group F*p of the field of integers modulo p. The algorithms and their analyses depend on many different parts of number theory. Number theory is considered the purest of all sciences, and within number theory the hunt for large primes and for factors of large numbers has always been remote from applications, even to other questions of a number-theoretic nature.