Choosing iteration maps for the parallel Pollard rho method

TL;DR

This paper analyzes the efficiency of parallel Pollard's rho method for integer factorization, providing asymptotic estimates and optimal parameter choices for different computational setups.

Contribution

It offers an asymptotic analysis of the expected running time based on parameter choices and proves that k=1 is optimal for single-machine scenarios without prior knowledge.

Findings

Optimal parameter k depends on the number of machines.

k=1 is optimal for single-machine factorization.

Provides asymptotic estimates for parallel rho method.

Abstract

Pollard's rho method finds a prime factor $p$ of an integer $N$ by searching for a collision in a map of the form $x \mapsto x^{2k} + c$ modulo $N$. This search can be parallelized to multiple machines, which may use distinct parameters $k$ and $c$. In this paper, we give an asymptotic estimate for the expected running time of the parallel rho method depending on the choice of $k$ for each machine. We also prove that $k = 1$ is the best choice for one machine, if nothing about $p$ is known in advance.