Spectral Analysis of Pollard Rho Collisions

TL;DR

This paper provides a rigorous spectral analysis of the Pollard rho algorithm, establishing an expected collision time close to the conjectured optimal and analyzing the mixing time of the associated random walk.

Contribution

It introduces a novel spectral analysis approach to estimate the mixing time and collision probability of the unaltered Pollard rho graph.

Findings

Expected collision time is O(√n (log n)^3)

Mixing time for the random walk is O((log n)^3)

Spectral analysis captures the effect of the squaring step

Abstract

We show that the classical Pollard rho algorithm for discrete logarithms produces a collision in expected time O(sqrt(n)(log n)^3). This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard rho graph, and is close to the conjectured optimal bound of O(sqrt(n)). The result is derived by showing that the mixing time for the random walk on this graph is O((log n)^3); without the squaring step in the Pollard rho algorithm, the mixing time would be exponential in log n. The technique involves a spectral analysis of directed graphs, which captures the effect of the squaring step.