Analysis of Pollard's Rho Factoring Method

Abstract

A comprehensive heuristic analysis of the Pollard's Rho Method (PRM) is given. The analysis is based on ultimate periods and tails distribution of sequences. If $n$ is the composite number to be factored, then an improved version of PRM is developed which has an expected run time of $O(\sqrt[8]{n}\ln n)$ on a multi-core architecture which utilized a clever way of evaluating polynomials.