2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2007)

Providence, Rhode Island

Oct. 21, 2007 to Oct. 23, 2007

ISSN: 0272-5428

ISBN: 0-7695-3010-9

pp: 215-223

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2007.38

ABSTRACT

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|).

INDEX TERMS

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CITATION

Prasad Tetali,
Ravi Montenegro,
Jeong Han Kim,
"Near Optimal Bounds for Collision in Pollard Rho for Discrete Log",

*2013 IEEE 54th Annual Symposium on Foundations of Computer Science*, vol. 00, no. , pp. 215-223, 2007, doi:10.1109/FOCS.2007.38