Non-Degeneracy of Pollard Rho Collisions

The Pollard ρ 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 bound for the collision time on a cyclic group of order n [4, 5]. 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 ρ algorithm successfully finds the discrete logarithm.