Using Equivalence Classes to Accelerate Solving the Discrete Logarithm Problem in a Short Interval

Abstract

The Pollard kangaroo method solves the discrete logarithm problem (DLP) in an interval of size N with heuristic average case expected running time approximately \(2 \sqrt{N}\) group operations. It is well-known that the Pollard rho method can be sped-up by using equivalence classes (such as orbits of points under an efficiently computed group homomorphism), but such ideas have not been used for the DLP in an interval. Indeed, it seems impossible to implement the standard kangaroo method with equivalence classes.

The main result of the paper is to give an algorithm, building on work of Gaudry and Schost, to solve the DLP in an interval of size N with heuristic average case expected running time of close to \(1.36\sqrt{N}\) group operations for groups with fast inversion. In practice the algorithm is not quite this fast, due to the usual problems with pseudorandom walks such as fruitless cycles. In addition, we present experimental results.