Abstract. In [Groups Complex. Cryptol. 3 (2011), 349–355] we showed that any hyperbolic limit group can be faithfully represented in . The proof was constructive in that given a fixed JSJ decomposition for the given limit group the representation can be constructed. The proof depended on showing that certain amalgams of groups admitting faithful representations into also admit such faithful representations. In this short note we give an elegant proof that the restriction to the hyperbolic case can be removed.
{"title":"Faithful representations of limit groups II","authors":"B. Fine, G. Rosenberger","doi":"10.1515/gcc-2013-0005","DOIUrl":"https://doi.org/10.1515/gcc-2013-0005","url":null,"abstract":"Abstract. In [Groups Complex. Cryptol. 3 (2011), 349–355] we showed that any hyperbolic limit group can be faithfully represented in . The proof was constructive in that given a fixed JSJ decomposition for the given limit group the representation can be constructed. The proof depended on showing that certain amalgams of groups admitting faithful representations into also admit such faithful representations. In this short note we give an elegant proof that the restriction to the hyperbolic case can be removed.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"195 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114972436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a cryptosystem which is complete for the class of probabilistic public-key cryptosystems with bounded error. Besides traditional encryption schemes such as RSA and El Gamal and probabilistic encryption of Goldwasser and Micali, this class contains also Ajtai-Dwork and NTRU cryptosystems. The latter two make errors with a small positive probability.
{"title":"A Complete Public-Key Cryptosystem","authors":"D. Grigoriev, E. Hirsch, Konstantin Pervyshev","doi":"10.1515/GCC.2009.1","DOIUrl":"https://doi.org/10.1515/GCC.2009.1","url":null,"abstract":"We present a cryptosystem which is complete for the class of probabilistic public-key cryptosystems with bounded error. Besides traditional encryption schemes such as RSA and El Gamal and probabilistic encryption of Goldwasser and Micali, this class contains also Ajtai-Dwork and NTRU cryptosystems. The latter two make errors with a small positive probability.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124095568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A generalized tetrahedron group is defined to be a group admitting the following presentation: , 2 ≤ l, m, n, p, q, r, where each Wi (a, b) is a cyclically reduced word involving both a and b. These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. In this paper, we build on previous work to show that the Tits alternative holds for Tsaranov's generalized tetrahedron groups, that is, if G is a Tsaranov generalized tetrahedron group then G contains a non-abelian free subgroup or is solvable-by-finite. The term Tits alternative comes from the respective property for finitely generated linear groups over a field (see [Tits, J. Algebra 20: 250–270, 1972]).
广义四面体群的定义是:2≤l, m, n, p, q, r，其中每一个Wi (A, b)是一个循环约简词，包含A和b。这些群出现在许多情况下，尤其是作为某些双曲轨道的基本群或作为广义三角形群的子群。本文在前人工作的基础上，证明了Tsaranov广义四面体群的Tits可选性，即如果G是Tsaranov广义四面体群，则G包含一个非阿贝耳自由子群或G是有限可解的。术语Tits替代来自于域上有限生成的线性群的各自性质(参见[Tits, J. Algebra 20: 250-270, 1972])。
{"title":"The Tits Alternative for Tsaranov's Generalized Tetrahedron Groups","authors":"V. G. Rebel, Miriam Hahn, G. Rosenberger","doi":"10.1515/GCC.2009.207","DOIUrl":"https://doi.org/10.1515/GCC.2009.207","url":null,"abstract":"A generalized tetrahedron group is defined to be a group admitting the following presentation: , 2 ≤ l, m, n, p, q, r, where each Wi (a, b) is a cyclically reduced word involving both a and b. These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. In this paper, we build on previous work to show that the Tits alternative holds for Tsaranov's generalized tetrahedron groups, that is, if G is a Tsaranov generalized tetrahedron group then G contains a non-abelian free subgroup or is solvable-by-finite. The term Tits alternative comes from the respective property for finitely generated linear groups over a field (see [Tits, J. Algebra 20: 250–270, 1972]).","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133337069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this note we give a straightforward proof that any hyperbolic limit group has a faithful representation in PSL(2, ℂ).
摘要本文给出了一个简单的证明，证明了任意双曲极限群在PSL(2，)中都有一个忠实的表示。
{"title":"A note on faithful representations of limit groups","authors":"B. Fine, G. Rosenberger","doi":"10.1515/gcc.2011.014","DOIUrl":"https://doi.org/10.1515/gcc.2011.014","url":null,"abstract":"Abstract In this note we give a straightforward proof that any hyperbolic limit group has a faithful representation in PSL(2, ℂ).","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115926028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we apply quantum algorithms to solve problems concerning fixed points and invariant subgroups of automorphisms. These efficient algorithms invoke a quantum algorithm which computes the intersection of multiple unsorted multisets whose elements originate from the same set. This intersection algorithm is an application of the Grover search procedure.
{"title":"Quantum algorithms for fixed points and invariant subgroups","authors":"M. Bonanome, S. Majewicz","doi":"10.1515/gcc.2011.013","DOIUrl":"https://doi.org/10.1515/gcc.2011.013","url":null,"abstract":"Abstract In this paper, we apply quantum algorithms to solve problems concerning fixed points and invariant subgroups of automorphisms. These efficient algorithms invoke a quantum algorithm which computes the intersection of multiple unsorted multisets whose elements originate from the same set. This intersection algorithm is an application of the Grover search procedure.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121394357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we modify the technique of cyclic permutations to work with the shifted conjugacy problem. We apply this technique to design a heuristic attack on the cryptographic authentication scheme based on shifted conjugacy of braids proposed by Dehornoy in [Using shifted conjugacy in braid-based cryptography, American Mathematical Society, 2006] and report experimental results.
在本文中，我们改进了循环置换技术来处理移位共轭问题。我们将该技术应用于Dehornoy在[Using shift - conjugacy in braid-based cryptography, American Mathematical Society, 2006]中提出的基于辫子移位共轭的密码认证方案设计了一种启发式攻击，并报告了实验结果。
{"title":"A Practical Attack on a Certain Braid Group Based Shifted Conjugacy Authentication Protocol","authors":"Jonathan Longrigg, A. Ushakov","doi":"10.1515/GCC.2009.275","DOIUrl":"https://doi.org/10.1515/GCC.2009.275","url":null,"abstract":"In this paper we modify the technique of cyclic permutations to work with the shifted conjugacy problem. We apply this technique to design a heuristic attack on the cryptographic authentication scheme based on shifted conjugacy of braids proposed by Dehornoy in [Using shifted conjugacy in braid-based cryptography, American Mathematical Society, 2006] and report experimental results.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121770016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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