Spectral analysis of Pollard Rho collisions

Abstract

We show that the classical Pollard p algorithm for discrete logarithms produces a collision in expected time O(√n(log n)3). This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard p graph, and is close to the conjectured optimal bound of O(√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 p 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. © Springer-Verlag Berlin Heidelberg 2006.