Abstract. We consider a key exchange procedure whose security is based on the difficulty of computing discrete logarithms in a group, and where exponentiation is hidden by a conjugation. We give a platform-dependent cryptanalysis of this protocol. Finally, to take full advantage of this procedure, we propose a group of matrices over a noncommutative ring as platform group.
{"title":"A Diffie–Hellman key exchange protocol using matrices over noncommutative rings","authors":"Mohammad Eftekhari","doi":"10.1515/gcc-2012-0001","DOIUrl":"https://doi.org/10.1515/gcc-2012-0001","url":null,"abstract":"Abstract. We consider a key exchange procedure whose security is based on the difficulty of computing discrete logarithms in a group, and where exponentiation is hidden by a conjugation. We give a platform-dependent cryptanalysis of this protocol. Finally, to take full advantage of this procedure, we propose a group of matrices over a noncommutative ring as platform group.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"104 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":"114661809","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 Algebraic attacks lead to the task of solving polynomial systems over 𝔽2. We study recent suggestions of using SAT-solvers for this task. In particular, we develop several strategies for converting the polynomial system to a set of CNF clauses. This generalizes the approach in [Bard, Courtois, Jefferson, Cryptology ePrint Archive 2007, 2007]. Moreover, we provide a novel way of transforming a system over 𝔽2 e to a (larger) system over 𝔽2. Finally, the efficiency of these methods is examined using standard examples such as CTC, DES, and Small Scale AES.
{"title":"Algebraic attacks using SAT-solvers","authors":"Philipp Jovanovic, M. Kreuzer","doi":"10.1515/gcc.2010.016","DOIUrl":"https://doi.org/10.1515/gcc.2010.016","url":null,"abstract":"Abstract Algebraic attacks lead to the task of solving polynomial systems over 𝔽2. We study recent suggestions of using SAT-solvers for this task. In particular, we develop several strategies for converting the polynomial system to a set of CNF clauses. This generalizes the approach in [Bard, Courtois, Jefferson, Cryptology ePrint Archive 2007, 2007]. Moreover, we provide a novel way of transforming a system over 𝔽2 e to a (larger) system over 𝔽2. Finally, the efficiency of these methods is examined using standard examples such as CTC, DES, and Small Scale AES.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"60 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":"116028208","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 [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.
{"title":"Existence and Non-Existence of Torsion in Maximal Arithmetic Fuchsian Groups","authors":"C. Maclachlan","doi":"10.1515/GCC.2009.287","DOIUrl":"https://doi.org/10.1515/GCC.2009.287","url":null,"abstract":"In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"48 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":"123619330","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 A Hurwitz group is any non-trivial finite quotient of the (2, 3, 7) triangle group, that is, any non-trivial finite group generated by elements x and y satisfying x 2 = y 3 = (xy)7 = 1. Every such group G is the conformal automorphism group of some compact Riemann surface of genus g > 1, with the property that |G| = 84(g – 1), which is the maximum possible order for given genus g. This paper provides an update on what is known about Hurwitz groups and related matters, following up the author's brief survey in Bull. Amer. Math. Soc.23 (1990).
{"title":"An update on Hurwitz groups","authors":"M. Conder","doi":"10.1515/gcc.2010.002","DOIUrl":"https://doi.org/10.1515/gcc.2010.002","url":null,"abstract":"Abstract A Hurwitz group is any non-trivial finite quotient of the (2, 3, 7) triangle group, that is, any non-trivial finite group generated by elements x and y satisfying x 2 = y 3 = (xy)7 = 1. Every such group G is the conformal automorphism group of some compact Riemann surface of genus g > 1, with the property that |G| = 84(g – 1), which is the maximum possible order for given genus g. This paper provides an update on what is known about Hurwitz groups and related matters, following up the author's brief survey in Bull. Amer. Math. Soc.23 (1990).","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"20 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":"128403319","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}
It is an open problem whether the shifted conjugacy (decision) problem in B ∞ is solvable. We settle this problem by reduction to an instance of the simultaneous conjugacy problem in Bn for some n ∈ ℕ.
{"title":"A Note on the Shifted Conjugacy Problem in Braid Groups","authors":"A. Kalka, E. Liberman, M. Teicher","doi":"10.1515/GCC.2009.227","DOIUrl":"https://doi.org/10.1515/GCC.2009.227","url":null,"abstract":"It is an open problem whether the shifted conjugacy (decision) problem in B ∞ is solvable. We settle this problem by reduction to an instance of the simultaneous conjugacy problem in Bn for some n ∈ ℕ.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"70 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":"130509467","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}
If R is a binomial ring, then a nilpotent R-powered group G is termed power-commutative if for any α ∈ R, [gα, h] = 1 implies [g, h] = 1 whenever gα ≠ 1. In this paper, we further contribute to the theory of nilpotent R-powered groups. In particular, we prove that if G is a nilpotent R-powered group of finite type which is not of finite π-type for any prime π ∈ R, then G is PC if and only if it is an abelian R-group.
{"title":"Power-Commutative Nilpotent R-Powered Groups","authors":"S. Majewicz, Marcos Zyman","doi":"10.1515/GCC.2009.297","DOIUrl":"https://doi.org/10.1515/GCC.2009.297","url":null,"abstract":"If R is a binomial ring, then a nilpotent R-powered group G is termed power-commutative if for any α ∈ R, [gα, h] = 1 implies [g, h] = 1 whenever gα ≠ 1. In this paper, we further contribute to the theory of nilpotent R-powered groups. In particular, we prove that if G is a nilpotent R-powered group of finite type which is not of finite π-type for any prime π ∈ R, then G is PC if and only if it is an abelian R-group.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"71 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":"124984510","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 classify the coordinate ℕ-monoids of algebraic sets over the additive monoid of natural numbers.
摘要本文对自然数加性单群上代数集的坐标_ -单群进行了分类。
{"title":"Algebraic geometry over natural numbers. The classification of coordinate monoids","authors":"A. Shevlyakov","doi":"10.1515/gcc.2010.007","DOIUrl":"https://doi.org/10.1515/gcc.2010.007","url":null,"abstract":"Abstract In this paper we classify the coordinate ℕ-monoids of algebraic sets over the additive monoid of natural numbers.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"10 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":"125301396","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 paper is a survey of known results (old and new) about equations over groups, mainly solvable groups. Current directions of research are discussed in some detail. A number of open questions are included.
{"title":"Equations over groups","authors":"V. Roman’kov","doi":"10.1515/gcc-2012-0015","DOIUrl":"https://doi.org/10.1515/gcc-2012-0015","url":null,"abstract":"Abstract. This paper is a survey of known results (old and new) about equations over groups, mainly solvable groups. Current directions of research are discussed in some detail. A number of open questions are included.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"298 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":"123127014","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 The Latin square is a good candidate in a secret sharing scheme to represent a secret, because of the huge number of the Latin squares for a reasonably large order. This makes outsiders difficult to discover the secret due to tremendous possibilities. We can improve the efficiency by distributing the shares of the critical set, instead of the full Latin square, to the participants. By different critical sets of the same Latin square, different secret sharing schemes can be implemented. However, finding a critical set of a large order Latin square is very difficult. This makes the implementation of Latin square based secret sharing scheme hard. We explore these limitations, then we propose to apply herding hash technique to overcome them.
{"title":"The Latin squares and the secret sharing schemes","authors":"Chi Sing Chum, Xiaowen Zhang","doi":"10.1515/gcc.2010.011","DOIUrl":"https://doi.org/10.1515/gcc.2010.011","url":null,"abstract":"Abstract The Latin square is a good candidate in a secret sharing scheme to represent a secret, because of the huge number of the Latin squares for a reasonably large order. This makes outsiders difficult to discover the secret due to tremendous possibilities. We can improve the efficiency by distributing the shares of the critical set, instead of the full Latin square, to the participants. By different critical sets of the same Latin square, different secret sharing schemes can be implemented. However, finding a critical set of a large order Latin square is very difficult. This makes the implementation of Latin square based secret sharing scheme hard. We explore these limitations, then we propose to apply herding hash technique to overcome them.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"9 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":"124071093","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}
After reviewing the importance of the Word Problem and the Conjugacy Problem for group-based cryptosystems, this paper offers an efficient method for solving both problems in the Shuffle Group.
{"title":"The Word and Conjugacy Problem for Shuffle Groups","authors":"Daniella Bak Shnaps","doi":"10.1515/GCC.2009.143","DOIUrl":"https://doi.org/10.1515/GCC.2009.143","url":null,"abstract":"After reviewing the importance of the Word Problem and the Conjugacy Problem for group-based cryptosystems, this paper offers an efficient method for solving both problems in the Shuffle Group.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"3 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":"121490538","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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