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Providence, Rhode Island
Oct. 21, 2007 to Oct. 23, 2007
ISBN: 0-7695-3010-9
pp: 215-223
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 {\rm O}(\sqrt {\left| G \right|\log \left| G \right|\log \log \left| G \right|} ) steps, not far from the widely conjectured value of \Theta (\sqrt {\left| G \right|} ). This improves upon a recent result of Miller-Venkatesan which showed an upper bound of {\rm O}(\sqrt {\left| G \right|} \log ^3 \left| G \right|). Our proof is based on analyzing an appropriate nonreversible, non-lazy random walk on a discrete cycle of (odd) length \left| G \right|, and showing that the mixing time of the corresponding walk is {\rm O}(\log \left| G \right|\log \log \left| G \right|).
Jeong Han Kim, Ravi Montenegro, Prasad Tetali, "Near Optimal Bounds for Collision in Pollard Rho for Discrete Log", FOCS, 2007, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2007, pp. 215-223, doi:10.1109/FOCS.2007.38
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