We propose an authentication scheme where forgery (a.k.a. impersonation) seems infeasible without finding the prover's long-term private key. The latter would follow from solving the conjugacy search problem in the platform (noncommutative) semigroup, i.e., to recovering X from X –1 AX and A. The platform semigroup that we suggest here is the semigroup of n × n matrices over truncated multivariable polynomials over a ring.
{"title":"Authentication from Matrix Conjugation","authors":"D. Grigoriev, V. Shpilrain","doi":"10.1515/GCC.2009.199","DOIUrl":"https://doi.org/10.1515/GCC.2009.199","url":null,"abstract":"We propose an authentication scheme where forgery (a.k.a. impersonation) seems infeasible without finding the prover's long-term private key. The latter would follow from solving the conjugacy search problem in the platform (noncommutative) semigroup, i.e., to recovering X from X –1 AX and A. The platform semigroup that we suggest here is the semigroup of n × n matrices over truncated multivariable polynomials over a ring.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134304656","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 Decision problems are problems of the following nature: given a property and an object , find out whether or not the object has the property . On the other hand, witness problems are: given a property and an object with the property , find a proof of the fact that indeed has the property . On the third hand(?!), search problems are of the following nature: given a property and an object with the property , find something “material” establishing the property ; for example, given two conjugate elements of a group, find a conjugator. In this survey our focus is on various search problems in group theory, including the word search problem, the subgroup membership search problem, the conjugacy search problem, and others.
{"title":"Search and witness problems in group theory","authors":"V. Shpilrain","doi":"10.1515/gcc.2010.015","DOIUrl":"https://doi.org/10.1515/gcc.2010.015","url":null,"abstract":"Abstract Decision problems are problems of the following nature: given a property and an object , find out whether or not the object has the property . On the other hand, witness problems are: given a property and an object with the property , find a proof of the fact that indeed has the property . On the third hand(?!), search problems are of the following nature: given a property and an object with the property , find something “material” establishing the property ; for example, given two conjugate elements of a group, find a conjugator. In this survey our focus is on various search problems in group theory, including the word search problem, the subgroup membership search problem, the conjugacy search problem, and others.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130510534","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 This is a survey paper on algorithms that have been developed during the last 25 years for the explicit computation of the structure of an associative algebra of finite dimension over either a finite field or an algebraic number field. This constructive approach was initiated in 1985 by Friedl and Rónyai and has since been developed by Cohen, de Graaf, Eberly, Giesbrecht, Ivanyos, Küronya and Wales. I illustrate these algorithms with the case n = 2 of the rational semigroup algebra of the partial transformation semigroup PTn on n elements; this generalizes the full transformation semigroup and the symmetric inverse semigroup, and these generalize the symmetric group Sn .
{"title":"How to compute the Wedderburn decomposition of a finite-dimensional associative algebra","authors":"M. Bremner","doi":"10.1515/gcc.2011.003","DOIUrl":"https://doi.org/10.1515/gcc.2011.003","url":null,"abstract":"Abstract This is a survey paper on algorithms that have been developed during the last 25 years for the explicit computation of the structure of an associative algebra of finite dimension over either a finite field or an algebraic number field. This constructive approach was initiated in 1985 by Friedl and Rónyai and has since been developed by Cohen, de Graaf, Eberly, Giesbrecht, Ivanyos, Küronya and Wales. I illustrate these algorithms with the case n = 2 of the rational semigroup algebra of the partial transformation semigroup PTn on n elements; this generalizes the full transformation semigroup and the symmetric inverse semigroup, and these generalize the symmetric group Sn .","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131779431","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 Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order . This answers a question of Benjamini. The same also holds when the generating set is taken to be a symmetric set of size 2k.
{"title":"The diameter of a random Cayley graph of ℤ q","authors":"Gideon Amir, O. Gurel-Gurevich","doi":"10.1515/gcc.2010.004","DOIUrl":"https://doi.org/10.1515/gcc.2010.004","url":null,"abstract":"Abstract Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order . This answers a question of Benjamini. The same also holds when the generating set is taken to be a symmetric set of size 2k.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"14 1-2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133036869","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 This is a short proof of the existence of finite sets of edges in graphs with more than one end, such that after removing them we obtain two components which are nested with all their isomorphic images. This was first done in “Cutting up graphs” [Dunwoody, Combinatorica 2: 15–23, 1982]. Together with a certain tree construction and some elementary Bass–Serre theory this yields a combinatorial proof of Stallings' theorem on the structure of finitely generated groups with more than one end.
{"title":"Cutting up graphs revisited – a short proof of Stallings' structure theorem","authors":"B. Krön","doi":"10.1515/gcc.2010.013","DOIUrl":"https://doi.org/10.1515/gcc.2010.013","url":null,"abstract":"Abstract This is a short proof of the existence of finite sets of edges in graphs with more than one end, such that after removing them we obtain two components which are nested with all their isomorphic images. This was first done in “Cutting up graphs” [Dunwoody, Combinatorica 2: 15–23, 1982]. Together with a certain tree construction and some elementary Bass–Serre theory this yields a combinatorial proof of Stallings' theorem on the structure of finitely generated groups with more than one end.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"120 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128017507","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 Richard Thompson's group F has a two generator presentation This paper studies when a pair of elements in F consists of the images of the generators x 0 and x 1 under a self monomorphism.
摘要:本文研究了F中的一对元素在自单态下由生成元x 0和x 1的象组成的情况。
{"title":"Subgroups of R. Thompson's group F that are isomorphic to F","authors":"B. Wassink","doi":"10.1515/gcc.2011.009","DOIUrl":"https://doi.org/10.1515/gcc.2011.009","url":null,"abstract":"Abstract Richard Thompson's group F has a two generator presentation This paper studies when a pair of elements in F consists of the images of the generators x 0 and x 1 under a self monomorphism.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128825780","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 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph, and we obtain new proofs of the mentioned results and others previously depending on these.
{"title":"On Cayley graphs of virtually free groups","authors":"Yago Antolín","doi":"10.1515/gcc.2011.012","DOIUrl":"https://doi.org/10.1515/gcc.2011.012","url":null,"abstract":"Abstract In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph, and we obtain new proofs of the mentioned results and others previously depending on these.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115456583","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}
The object of this paper is to give a proof of the following theorem: S/P ≅ Λ mn ⊆ ℚ+, where S/P is a certain torsion-free factor group of the Baumslag-Solitar group 〈a, b; a –1 bma = bn | m ≠ 0, n ≠ 0, m, n ∈ ℤ〉, with m and n are relatively prime, and Λ mn is a subgroup of the additive group of the rational numbers ℚ+.
{"title":"Torsion-free Abelian Factor Groups of the Baumslag-Solitar Groups and Subgroups of the Additive Group of the Rational Numbers","authors":"A. Clement","doi":"10.1515/GCC.2009.165","DOIUrl":"https://doi.org/10.1515/GCC.2009.165","url":null,"abstract":"The object of this paper is to give a proof of the following theorem: S/P ≅ Λ mn ⊆ ℚ+, where S/P is a certain torsion-free factor group of the Baumslag-Solitar group 〈a, b; a –1 bma = bn | m ≠ 0, n ≠ 0, m, n ∈ ℤ〉, with m and n are relatively prime, and Λ mn is a subgroup of the additive group of the rational numbers ℚ+.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133731430","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 We construct a short presentation of the ring of n × n matrices over Z with only 2 generators and 3 relations.
摘要我们构造了Z上n × n矩阵环的一个简短表示，它只有2个生成子和3个关系。
{"title":"Presentations of matrix rings","authors":"M. Kassabov","doi":"10.1515/gcc.2010.003","DOIUrl":"https://doi.org/10.1515/gcc.2010.003","url":null,"abstract":"Abstract We construct a short presentation of the ring of n × n matrices over Z with only 2 generators and 3 relations.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123141894","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 We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al. are polynomial-time reducible to each other. We study two new geodesic problems which arise in a previous paper of the authors and Fusy.
{"title":"Some geodesic problems in groups","authors":"M. Elder, A. Rechnitzer","doi":"10.1515/gcc.2010.014","DOIUrl":"https://doi.org/10.1515/gcc.2010.014","url":null,"abstract":"Abstract We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al. are polynomial-time reducible to each other. We study two new geodesic problems which arise in a previous paper of the authors and Fusy.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"863 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117143071","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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