**Collision Bounds for the Additive Pollard Rho Algorithm for Solving Discrete Logarithms**

*Joppe W. Bos and Alina Dudeanu and Dimitar Jetchev*

**Abstract: **We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm problem in a general cyclic group $G$. Unlike the setting studied by Kim et al. we consider additive walks: the setting used in practice to solve the elliptic curve discrete logarithm problem. Our bounds differ from the birthday bound $O(\sqrt{|G|})$ by a factor of $\sqrt{\log{|G|}}$ and are based on mixing time estimates for random walks on finite abelian groups due to Hildebrand.

**Category / Keywords: **Pollard rho, additive walk, collision bound, random walk, mixing times

**Date: **received 23 Feb 2012

**Contact author: **joppe bos at epfl ch

**Available format(s): **PDF | BibTeX Citation

**Version: **20120223:215243 (All versions of this report)

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