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 of log||$\sqrt{\log {\vert \mathbf {G}\vert }}$ and are based on mixing time estimates for random walks on finite abelian groups due to Dou and Hildebrand
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