Non-degeneracy of Pollard Rho Collisions

Abstract

The Pollard Rho algorithm is a widely used algorithm for solving discrete logarithms on general cyclic groups, including elliptic curves. Recently the first nontrivial runtime estimates were provided for it, culminating in a sharp O(sqrt(n)) bound for the collision time on a cyclic group of order n. In this paper we show that for n satisfying a mild arithmetic condition, the collisions guaranteed by these results are nondegenerate with high probability: that is, the Pollard Rho algorithm successfully finds the discrete logarithm.