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Near Optimal Bounds for Collision in Pollard Rho for Discrete Log
 Proc. of the 48th Annual Symposium on Foundations of Computer Science (FOCS
, 2007
"... We analyze a fairly standard idealization of Pollard’s Rho algorithm for finding the discrete logarithm in a cyclic group G. It is found that, with high probability, a collision occurs in O ( � G  log G  log log G) steps, not far from the widely conjectured value of Θ ( � G). This improves ..."
Abstract

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We analyze a fairly standard idealization of Pollard’s Rho algorithm for finding the discrete logarithm in a cyclic group G. It is found that, with high probability, a collision occurs in O ( � G  log G  log log G) steps, not far from the widely conjectured value of Θ ( � G). This improves upon a recent result of Miller–Venkatesan which showed an upper bound of O ( � G  log 3 G). Our proof is based on analyzing an appropriate nonreversible, nonlazy random walk on a discrete cycle of (odd) length G, and showing that the mixing time of the corresponding walk is O(log G  log log G). 1