{"id":357917,"date":"2017-01-25T14:32:52","date_gmt":"2017-01-25T22:32:52","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&p=357917"},"modified":"2018-10-16T20:02:09","modified_gmt":"2018-10-17T03:02:09","slug":"birthday-paradox-markov-chains-optimal-bound-collision-pollard-rho-algorithm-discrete-logarithm","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/birthday-paradox-markov-chains-optimal-bound-collision-pollard-rho-algorithm-discrete-logarithm\/","title":{"rendered":"A Birthday Paradox For Markov Chains With An Optimal Bound For Collision In The Pollard Rho Algorithm For Discrete Logarithm"},"content":{"rendered":"

We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard’s Rho algorithm for finding the discrete logarithm in a cyclic group G<\/span><\/span><\/span><\/span> and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in \u0398<\/span>(<\/span>|<\/span><\/span><\/span>G<\/span>|<\/span><\/span><\/span><\/span>\u2212\u2212\u2212\u221a<\/span>)<\/span><\/span><\/span><\/span> steps. Moreover, for the parallelized distinguished points algorithm on J<\/span><\/span><\/span><\/span> processors we find that \u0398<\/span>(<\/span>|<\/span><\/span><\/span>G<\/span>|<\/span><\/span><\/span><\/span>\u2212\u2212\u2212\u221a<\/span>\/<\/span><\/span><\/span>J<\/span>)<\/span><\/span><\/span><\/span> steps suffices. These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from G<\/span><\/span><\/span><\/span>.<\/p>\n","protected":false},"excerpt":{"rendered":"

We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard’s Rho algorithm for finding the discrete logarithm in a cyclic group G and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Institute of Mathematical Statistics","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"The Annals of Applied 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