{"title":"离散对数波拉德碰撞的近最优界","authors":"J. Kim, R. Montenegro, P. Tetali","doi":"10.1109/FOCS.2007.44","DOIUrl":null,"url":null,"abstract":"We analyze-a fairly standard idealization of Pollard's rho algorithm for finding the discrete logarithm in acyclic group G. It is found that, with high probability, a collision occurs in O(radic( |G|log|G|log log|G|)) steps, not far from the widely conjectured value of Theta(radic|G|). Tins improves upon a recent result of Miller-Venkalesan which showed an upper bound of O(radic|G|log3|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 O(log|G|log log|G|).","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"27","resultStr":"{\"title\":\"Near Optimal Bounds for Collision in Pollard Rho for Discrete Log\",\"authors\":\"J. Kim, R. Montenegro, P. Tetali\",\"doi\":\"10.1109/FOCS.2007.44\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We analyze-a fairly standard idealization of Pollard's rho algorithm for finding the discrete logarithm in acyclic group G. It is found that, with high probability, a collision occurs in O(radic( |G|log|G|log log|G|)) steps, not far from the widely conjectured value of Theta(radic|G|). Tins improves upon a recent result of Miller-Venkalesan which showed an upper bound of O(radic|G|log3|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 O(log|G|log log|G|).\",\"PeriodicalId\":197431,\"journal\":{\"name\":\"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2007-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"27\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/FOCS.2007.44\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/FOCS.2007.44","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Near Optimal Bounds for Collision in Pollard Rho for Discrete Log
We analyze-a fairly standard idealization of Pollard's rho algorithm for finding the discrete logarithm in acyclic group G. It is found that, with high probability, a collision occurs in O(radic( |G|log|G|log log|G|)) steps, not far from the widely conjectured value of Theta(radic|G|). Tins improves upon a recent result of Miller-Venkalesan which showed an upper bound of O(radic|G|log3|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 O(log|G|log log|G|).