{"paper":{"title":"Near Optimal Bounds for Collision in Pollard Rho for Discrete Log","license":"","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.NT","authors_text":"Jeong Han Kim, Prasad Tetali, Ravi Montenegro","submitted_at":"2006-11-19T22:02:06Z","abstract_excerpt":"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 $O(\\sqrt{|G|\\log |G| \\log \\log |G|})$ steps, not far from the widely conjectured value of $\\Theta(\\sqrt{|G|})$. This improves upon a recent result of Miller--Venkatesan which showed an upper bound of $O(\\sqrt{|G|}\\log^3 |G|)$. Our proof is based on analyzing an appropriate nonreversible, non-lazy random walk on a discrete cycle of (odd) length |G|, and showing that the mixing time of the corresponding walk is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611586","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}