Abstract
We show that the classical Pollard ρ algorithm for discrete logarithms produces a collision in expected time \(O(\sqrt{n}(\log n)^3)\). This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard ρ graph, and is close to the conjectured optimal bound of \(O(\sqrt{n})\). The result is derived by showing that the mixing time for the random walk on this graph is O((logn)3); without the squaring step in the Pollard ρ algorithm, the mixing time would be exponential in logn. The technique involves a spectral analysis of directed graphs, which captures the effect of the squaring step.
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Miller, S.D., Venkatesan, R. (2006). Spectral Analysis of Pollard Rho Collisions. In: Hess, F., Pauli, S., Pohst, M. (eds) Algorithmic Number Theory. ANTS 2006. Lecture Notes in Computer Science, vol 4076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11792086_40
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DOI: https://doi.org/10.1007/11792086_40
Publisher Name: Springer, Berlin, Heidelberg
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