Abstract The discrete logarithm problem is one of the backbones in public key cryptography. In this paper we study the discrete logarithm problem in the group of circulant matrices over a finite field.
摘要离散对数问题是公钥密码学的主干问题之一。本文研究有限域上循环矩阵群的离散对数问题。
{"title":"The discrete logarithm problem in the group of non-singular circulant matrices","authors":"A. Mahalanobis","doi":"10.1515/gcc.2010.006","DOIUrl":"https://doi.org/10.1515/gcc.2010.006","url":null,"abstract":"Abstract The discrete logarithm problem is one of the backbones in public key cryptography. In this paper we study the discrete logarithm problem in the group of circulant matrices over a finite field.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115609801","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 (graph-)group-valued random element ξ, discuss the properties of a mean-set 𝔼(ξ), and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for ξ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.
{"title":"Strong law of large numbers on graphs and groups","authors":"N. Mosina, A. Ushakov","doi":"10.1515/gcc.2011.004","DOIUrl":"https://doi.org/10.1515/gcc.2011.004","url":null,"abstract":"Abstract We consider (graph-)group-valued random element ξ, discuss the properties of a mean-set 𝔼(ξ), and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for ξ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130384738","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 show that the universal theory of torsion groups is strongly contained in the universal theory of finite groups. This answers a question of Dyson. We also prove that the universal theory of some natural classes of torsion groups is undecidable. Finally we observe that the universal theory of the class of hyperbolic groups is undecidable and use this observation to construct a lacunary hyperbolic group with undecidable universal theory. Surprisingly, torsion groups play an important role in the proof of the latter results.
{"title":"On the Universal Theory of Torsion and Lacunary Hyperbolic Groups","authors":"D. Osin","doi":"10.1515/GCC.2009.311","DOIUrl":"https://doi.org/10.1515/GCC.2009.311","url":null,"abstract":"We show that the universal theory of torsion groups is strongly contained in the universal theory of finite groups. This answers a question of Dyson. We also prove that the universal theory of some natural classes of torsion groups is undecidable. Finally we observe that the universal theory of the class of hyperbolic groups is undecidable and use this observation to construct a lacunary hyperbolic group with undecidable universal theory. Surprisingly, torsion groups play an important role in the proof of the latter results.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125464890","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 Shephard groups are common extensions of Artin and Coxeter groups. They appear, for example, in algebraic study of manifolds. An infinite family of Shephard groups which are not Artin or Coxeter groups is considered. Using techniques form small cancellation theory we show that the groups in this family are bi-automatic.
{"title":"On Shephard groups with large triangles","authors":"Uri Weiss","doi":"10.1515/GCC.2010.001","DOIUrl":"https://doi.org/10.1515/GCC.2010.001","url":null,"abstract":"Abstract Shephard groups are common extensions of Artin and Coxeter groups. They appear, for example, in algebraic study of manifolds. An infinite family of Shephard groups which are not Artin or Coxeter groups is considered. Using techniques form small cancellation theory we show that the groups in this family are bi-automatic.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114960009","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 prove that for any finite Thurston-type ordering < T on the braid group Bn , the restriction to the positive braid monoid (, < T ) is a well-ordered set of order type ω ω n–2 . The proof uses a combinatorial description of the ordering < T . Our combinatorial description is based on a new normal form for positive braids which we call the (-normal form. It can be seen as a generalization of Burckel's normal form and Dehornoy's Φ-normal form (alternating normal form).
{"title":"On finite Thurston-type orderings of braid groups","authors":"Tetsuya Ito","doi":"10.1515/GCC.2010.009","DOIUrl":"https://doi.org/10.1515/GCC.2010.009","url":null,"abstract":"Abstract We prove that for any finite Thurston-type ordering < T on the braid group Bn , the restriction to the positive braid monoid (, < T ) is a well-ordered set of order type ω ω n–2 . The proof uses a combinatorial description of the ordering < T . Our combinatorial description is based on a new normal form for positive braids which we call the (-normal form. It can be seen as a generalization of Burckel's normal form and Dehornoy's Φ-normal form (alternating normal form).","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129636646","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, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system ℜ that satisfies the condition that each rule in ℜ with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from ℜ embeds in the group. As an example, we give a simple proof that right angled Artin monoids embed in the corresponding right angled Artin groups. This is a special case of the well-known result of Paris that Artin monoids embed in their groups.
{"title":"Rewriting Systems and Embedding of Monoids in Groups","authors":"Fabienne Chouraqui","doi":"10.1515/GCC.2009.131","DOIUrl":"https://doi.org/10.1515/GCC.2009.131","url":null,"abstract":"In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system ℜ that satisfies the condition that each rule in ℜ with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from ℜ embeds in the group. As an example, we give a simple proof that right angled Artin monoids embed in the corresponding right angled Artin groups. This is a special case of the well-known result of Paris that Artin monoids embed in their groups.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123427362","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 goal of this paper is to introduce ideas and methodology of the generic case complexity to cryptography community. This relatively new approach allows one to analyze the behavior of an algorithm on “most” inputs in a simple and intuitive fashion which has some practical advantages over classical methods based on averaging. We present an alternative definition of one-way function using the concepts of generic case complexity and show its equivalence to the standard definition. In addition we demonstrate the convenience of the new approach by giving a short proof that extending adversaries to a larger class of partial algorithms with errors does not change the strength of the security assumption.
{"title":"Generic Case Complexity and One-Way Functions","authors":"A. Myasnikov","doi":"10.1515/GCC.2009.13","DOIUrl":"https://doi.org/10.1515/GCC.2009.13","url":null,"abstract":"The goal of this paper is to introduce ideas and methodology of the generic case complexity to cryptography community. This relatively new approach allows one to analyze the behavior of an algorithm on “most” inputs in a simple and intuitive fashion which has some practical advantages over classical methods based on averaging. We present an alternative definition of one-way function using the concepts of generic case complexity and show its equivalence to the standard definition. In addition we demonstrate the convenience of the new approach by giving a short proof that extending adversaries to a larger class of partial algorithms with errors does not change the strength of the security assumption.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126735142","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 Anshel-Anshel-Goldfeld-Lemieux (abbreviated AAGL) key agreement protocol [Contemp. Math. 418: 1–34, 2006] is proposed to be used on low-cost platforms which constraint the use of computational resources. The core of the protocol is the concept of an Algebraic Eraser TM (abbreviated AE) which is claimed to be a suitable primitive for use within lightweight cryptography. The AE primitive is based on a new and ingenious idea of using an action of a semidirect product on a (semi)group to obscure involved algebraic structures. The underlying motivation for AAGL protocol is the need to secure networks which deploy Radio Frequency Identification (RFID) tags used for identification, authentication, tracing and point-of-sale applications. In this paper we revisit the computational problem on which AE relies and heuristically analyze its hardness. We show that for proposed parameter values it is impossible to instantiate a secure protocol. To be more precise, in 100% of randomly generated instances of the protocol we were able to find a secret conjugator z generated by the TTP algorithm (part of AAGL protocol).
{"title":"Cryptanalysis of the Anshel-Anshel-Goldfeld-Lemieux Key Agreement Protocol","authors":"A. Myasnikov, A. Ushakov","doi":"10.1515/GCC.2009.63","DOIUrl":"https://doi.org/10.1515/GCC.2009.63","url":null,"abstract":"The Anshel-Anshel-Goldfeld-Lemieux (abbreviated AAGL) key agreement protocol [Contemp. Math. 418: 1–34, 2006] is proposed to be used on low-cost platforms which constraint the use of computational resources. The core of the protocol is the concept of an Algebraic Eraser TM (abbreviated AE) which is claimed to be a suitable primitive for use within lightweight cryptography. The AE primitive is based on a new and ingenious idea of using an action of a semidirect product on a (semi)group to obscure involved algebraic structures. The underlying motivation for AAGL protocol is the need to secure networks which deploy Radio Frequency Identification (RFID) tags used for identification, authentication, tracing and point-of-sale applications. In this paper we revisit the computational problem on which AE relies and heuristically analyze its hardness. We show that for proposed parameter values it is impossible to instantiate a secure protocol. To be more precise, in 100% of randomly generated instances of the protocol we were able to find a secret conjugator z generated by the TTP algorithm (part of AAGL protocol).","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131494937","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}
This article surveys many standard results about the braid group, with emphasis on simplifying the usual algebraic proofs. We use van der Waerden's trick to illuminate the Artin-Magnus proof of the classic presentation of the braid group considered as the algebraic mapping-class group of a disc with punctures. We give a simple, new proof of the σ 1-trichotomy for the braid group, and, hence, recover the Dehornoy right-ordering of the braid group. We give three proofs of the Birman-Hilden theorem concerning the fidelity of braid-group actions on free products of finite cyclic groups, and discuss the consequences derived by Perron-Vannier and the connections with Artin groups and the Wada representations. The first, very direct, proof, is due to Crisp-Paris and uses the σ 1-trichotomy and the Larue-Shpilrain technique. The second proof arises by studying ends of free groups, and gives interesting extra information. The third proof arises from Larue's study of polygonal curves in discs with punctures, and gives extremely detailed information.
本文综述了关于辫群的许多标准结果，重点对常用的代数证明进行了简化。我们使用van der Waerden的技巧来阐明辫群作为带穿孔圆盘的代数映射类群的经典表示的Artin-Magnus证明。给出了辫群的σ 1-三分性的一个简单的新证明，从而恢复了辫群的Dehornoy右序性。给出了有限循环群自由积上关于辫群作用保真度的Birman-Hilden定理的三个证明，讨论了Perron-Vannier定理的结论及其与Artin群和Wada表示的联系。第一个非常直接的证明是由Crisp-Paris提出的，它使用了σ 1-三分法和Larue-Shpilrain技术。第二种证明是通过研究自由群的端点产生的，它提供了有趣的额外信息。第三个证明来自于Larue对带有穿孔的圆盘的多边形曲线的研究，并且给出了非常详细的信息。
{"title":"Actions of the Braid Group, and New Algebraic Proofs of Results of Dehornoy and Larue","authors":"Lluís Bacardit, Warren Dicks","doi":"10.1515/GCC.2009.77","DOIUrl":"https://doi.org/10.1515/GCC.2009.77","url":null,"abstract":"This article surveys many standard results about the braid group, with emphasis on simplifying the usual algebraic proofs. We use van der Waerden's trick to illuminate the Artin-Magnus proof of the classic presentation of the braid group considered as the algebraic mapping-class group of a disc with punctures. We give a simple, new proof of the σ 1-trichotomy for the braid group, and, hence, recover the Dehornoy right-ordering of the braid group. We give three proofs of the Birman-Hilden theorem concerning the fidelity of braid-group actions on free products of finite cyclic groups, and discuss the consequences derived by Perron-Vannier and the connections with Artin groups and the Wada representations. The first, very direct, proof, is due to Crisp-Paris and uses the σ 1-trichotomy and the Larue-Shpilrain technique. The second proof arises by studying ends of free groups, and gives interesting extra information. The third proof arises from Larue's study of polygonal curves in discs with punctures, and gives extremely detailed information.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130416099","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}
There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property and the information that there are objects with the property , find at least one particular object with the property . So far, no cryptographic protocol based on a search problem in a non-commutative (semi)group has been recognized as secure enough to be a viable alternative to established protocols (such as RSA) based on commutative (semi)groups, although most of these protocols are more efficient than RSA is. In this paper, we suggest to use decision problems from combinatorial group theory as the core of a public key establishment protocol or a public key cryptosystem. Decision problems are problems of the following nature: given a property and an object , find out whether or not the object has the property . By using a popular decision problem, the word problem, we design a cryptosystem with the following features: (1) Bob transmits to Alice an encrypted binary sequence which Alice decrypts correctly with probability “very close” to 1; (2) the adversary, Eve, who is granted arbitrarily high (but fixed) computational speed, cannot positively identify (at least, in theory), by using a “brute force attack”, the “1” or “0” bits in Bob's binary sequence. In other words: no matter what computational speed we grant Eve at the outset, there is no guarantee that her “brute force attack” program will give a conclusive answer (or an answer which is correct with overwhelming probability) about any bit in Bob's sequence.
{"title":"Using Decision Problems in Public Key Cryptography","authors":"V. Shpilrain, Gabriel Zapata","doi":"10.1515/GCC.2009.33","DOIUrl":"https://doi.org/10.1515/GCC.2009.33","url":null,"abstract":"There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property and the information that there are objects with the property , find at least one particular object with the property . So far, no cryptographic protocol based on a search problem in a non-commutative (semi)group has been recognized as secure enough to be a viable alternative to established protocols (such as RSA) based on commutative (semi)groups, although most of these protocols are more efficient than RSA is. In this paper, we suggest to use decision problems from combinatorial group theory as the core of a public key establishment protocol or a public key cryptosystem. Decision problems are problems of the following nature: given a property and an object , find out whether or not the object has the property . By using a popular decision problem, the word problem, we design a cryptosystem with the following features: (1) Bob transmits to Alice an encrypted binary sequence which Alice decrypts correctly with probability “very close” to 1; (2) the adversary, Eve, who is granted arbitrarily high (but fixed) computational speed, cannot positively identify (at least, in theory), by using a “brute force attack”, the “1” or “0” bits in Bob's binary sequence. In other words: no matter what computational speed we grant Eve at the outset, there is no guarantee that her “brute force attack” program will give a conclusive answer (or an answer which is correct with overwhelming probability) about any bit in Bob's sequence.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124886295","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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