Kruskal’s Principle and Collision Time for Monotone Transitive Walks on the Integers

A set of positive generators S ⊂ N with a probability distribution p on S induces a monotone transitive walk on the integers Z, with P(x, x+s) = p(s). Kruskal’s principle observes that when two independent copies of the walk are started from nearby states then, with high probability, they do not have to travel far before visiting a common state (“collision distance”). We develop tools for determining the expected collision distance and the probability of collision within a certain distance. We then derive bounds in terms of “collision time”; These are used to prove that Pollard’s Kangaroo method solves the discrete logarithm problem g = h on a cyclic group in expected time (2 + o(1)) √ N , when x is in an interval [a, b] of size N = b − a + 1. We also resolve a conjecture of Pollard’s by showing that the same bound holds, when step sizes are generalized from powers of 2 to powers of any fixed n.