Non-uniform birthday problem revisited: Refined analysis and applications to discrete logarithms

Abstract

The result of birthday problem is of fundamental importance and has many applications in diverse areas, analyzing the discrete logarithm problem in particular. The birthday problem can be modeled as sampling balls with replacement until one is sampled twice. In the classical scenario, balls are sampled uniformly. Inspired by the analysis of algorithms solving certain discrete logarithms, Galbraith and Holmes considered the non-uniform birthday problem and gave an estimate on the number of sampling before having a collision up to the dominating term. In this paper, we give a refined analysis to the non-uniform birthday problem, and apply it to the analysis of an algorithm solving certain discrete logarithm problem. Experiment results suggest that parameter selection following this refined analysis performs better than related work.