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We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm problem in a general cyclic group   $\mathbf {G}$   . 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  (    |    |    )    $\mathcal {O}(\sqrt{\vert \mathbf {G}\vert })$    by a factor oflog   |    |$\sqrt{\log {\vert \mathbf {G}\vert }}$    and are based on mixing time estimates for random walks on finite abelian groups due to Dou and Hildebrand.

Collision bounds for the additive Pollard rho algorithm for solving discrete logarithms