{"title":"Baby-Step Giant-Step Algorithms for the Symmetric Group","authors":"E. Bach, Bryce Sandlund","doi":"10.1145/2930889.2930930","DOIUrl":null,"url":null,"abstract":"We study discrete logarithms in the setting of group actions. Suppose that G is a group that acts on a set S. When r and s are elements of S, a solution g to rg = s can be thought of as a kind of logarithm. In this paper, we study the case where G = Sn, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two subsets A and B of Sn, such that every permutation in Sn can be written as a product ab of elements from A and B. Our deterministic procedure is close to optimal, in the sense that A and B can be computed efficiently and |A| and |B| are not too far from sqrt(n!) in size. We also analyze randomized \"collision\" algorithms for the same problem.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"54 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2930889.2930930","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We study discrete logarithms in the setting of group actions. Suppose that G is a group that acts on a set S. When r and s are elements of S, a solution g to rg = s can be thought of as a kind of logarithm. In this paper, we study the case where G = Sn, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two subsets A and B of Sn, such that every permutation in Sn can be written as a product ab of elements from A and B. Our deterministic procedure is close to optimal, in the sense that A and B can be computed efficiently and |A| and |B| are not too far from sqrt(n!) in size. We also analyze randomized "collision" algorithms for the same problem.
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