Abstract. A subgroup H of a free group F is called inert in F if for every . In this paper we expand the known families of inert subgroups. We show that the inertia property holds for 1-generator endomorphisms. Equivalently, echelon subgroups in free groups are inert. An echelon subgroup is defined through a set of generators that are in echelon form with respect to some ordered basis of the free group, and may be seen as a generalization of a free factor. For example, the fixed subgroups of automorphisms of finitely generated free groups are echelon subgroups. The proofs follow mostly a graph-theoretic or combinatorial approach.
{"title":"On the intersection of subgroups in free groups: Echelon subgroups are inert","authors":"A. Rosenmann","doi":"10.1515/gcc-2013-0013","DOIUrl":"https://doi.org/10.1515/gcc-2013-0013","url":null,"abstract":"Abstract. A subgroup H of a free group F is called inert in F if for every . In this paper we expand the known families of inert subgroups. We show that the inertia property holds for 1-generator endomorphisms. Equivalently, echelon subgroups in free groups are inert. An echelon subgroup is defined through a set of generators that are in echelon form with respect to some ordered basis of the free group, and may be seen as a generalization of a free factor. For example, the fixed subgroups of automorphisms of finitely generated free groups are echelon subgroups. The proofs follow mostly a graph-theoretic or combinatorial approach.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121088511","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. It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into for an appropriate ; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in [J. Algebra 300 (2006), 376–383]. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented in [J. Algebra 300 (2006), 376–383].
{"title":"On the dimension of matrix representations of finitely generated torsion free nilpotent groups","authors":"Maggie E. Habeeb, Delaram Kahrobaei","doi":"10.1515/gcc-2013-0011","DOIUrl":"https://doi.org/10.1515/gcc-2013-0011","url":null,"abstract":"Abstract. It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into for an appropriate ; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in [J. Algebra 300 (2006), 376–383]. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented in [J. Algebra 300 (2006), 376–383].","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115617886","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 method for non-abelian Cramer-Shoup cryptosystem is presented. The role of decision and search is explored, and the platform of solvable / polycyclic group is suggested. In the process we review recent progress in non-abelian cryptography and post some open problems that naturally arise from this path of research.
{"title":"Decision and Search in Non-Abelian Cramer-Shoup Public Key Cryptosystem","authors":"Delaram Kahrobaei, M. Anshel","doi":"10.1515/GCC.2009.217","DOIUrl":"https://doi.org/10.1515/GCC.2009.217","url":null,"abstract":"A method for non-abelian Cramer-Shoup cryptosystem is presented. The role of decision and search is explored, and the platform of solvable / polycyclic group is suggested. In the process we review recent progress in non-abelian cryptography and post some open problems that naturally arise from this path of research.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134514725","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 non-associative key establishment protocols for all left self-distributive (LD), multi-LD-, and other left distributive systems. Instantiations of these protocols using generalized shifted conjugacy in braid groups lead to instances of a natural and apparently new group-theoretic problem, which we call the (subgroup) conjugacy coset problem.
{"title":"Non-associative key establishment for left distributive systems","authors":"A. Kalka, M. Teicher","doi":"10.1515/gcc-2013-0009","DOIUrl":"https://doi.org/10.1515/gcc-2013-0009","url":null,"abstract":"Abstract. We construct non-associative key establishment protocols for all left self-distributive (LD), multi-LD-, and other left distributive systems. Instantiations of these protocols using generalized shifted conjugacy in braid groups lead to instances of a natural and apparently new group-theoretic problem, which we call the (subgroup) conjugacy coset problem.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129556797","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 generic-case approach to algorithmic problems was suggested by Myasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies the behavior of an algorithm on “most” or “typical” inputs. The remaining inputs form the so-called black hole of the algorithm. In the present paper we consider Hilbert's tenth problem and use arithmetic circuits for the representation of Diophantine equations. We prove that this Diophantine problem is generically hard in the following sense. For every generic polynomial algorithm deciding this problem, there exists a polynomial algorithm for random generation of inputs from the black hole.
{"title":"Generic complexity of the Diophantine problem","authors":"A. Rybalov","doi":"10.1515/gcc-2013-0004","DOIUrl":"https://doi.org/10.1515/gcc-2013-0004","url":null,"abstract":"Abstract. The generic-case approach to algorithmic problems was suggested by Myasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies the behavior of an algorithm on “most” or “typical” inputs. The remaining inputs form the so-called black hole of the algorithm. In the present paper we consider Hilbert's tenth problem and use arithmetic circuits for the representation of Diophantine equations. We prove that this Diophantine problem is generically hard in the following sense. For every generic polynomial algorithm deciding this problem, there exists a polynomial algorithm for random generation of inputs from the black hole.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121458586","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 extend results of [Proc. Lond. Math. Soc. 104 (2012), 486–512] and prove shortlex automaticity and regularity of geodesics in a family of Artin groups that includes all groups of large type but also allows some commuting pairs of generators.
{"title":"Shortlex automaticity and geodesic regularity in Artin groups","authors":"D. Holt, Sarah Rees","doi":"10.1515/gcc-2013-0001","DOIUrl":"https://doi.org/10.1515/gcc-2013-0001","url":null,"abstract":"Abstract. We extend results of [Proc. Lond. Math. Soc. 104 (2012), 486–512] and prove shortlex automaticity and regularity of geodesics in a family of Artin groups that includes all groups of large type but also allows some commuting pairs of generators.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130821705","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}
Delaram Kahrobaei, Charalambos Koupparis, V. Shpilrain
Abstract. We offer a public key exchange protocol in the spirit of Diffie–Hellman, but we use (small) matrices over a group ring of a (small) symmetric group as the platform. This “nested structure” of the platform makes computation very efficient for legitimate parties. We discuss security of this scheme by addressing the Decision Diffie–Hellman (DDH) and Computational Diffie–Hellman (CDH) problems for our platform.
{"title":"Public key exchange using matrices over group rings","authors":"Delaram Kahrobaei, Charalambos Koupparis, V. Shpilrain","doi":"10.1515/gcc-2013-0007","DOIUrl":"https://doi.org/10.1515/gcc-2013-0007","url":null,"abstract":"Abstract. We offer a public key exchange protocol in the spirit of Diffie–Hellman, but we use (small) matrices over a group ring of a (small) symmetric group as the platform. This “nested structure” of the platform makes computation very efficient for legitimate parties. We discuss security of this scheme by addressing the Decision Diffie–Hellman (DDH) and Computational Diffie–Hellman (CDH) problems for our platform.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"145 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123304972","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 show that some problems in information security can be solved without using one-way functions. The latter are usually regarded as a central concept of cryptography, but the very existence of one-way functions depends on difficult conjectures in complexity theory, most notably on the notorious “” conjecture. This is why cryptographic primitives that do not employ one-way functions are often called “unconditionally secure”. In this paper, we suggest protocols for secure computation of the sum, product, and some other functions of two or more elements of an arbitrary constructible ring, without using any one-way functions. A new input that we offer here is that, in contrast with other proposals, we conceal “intermediate results” of a computation. For example, when we compute the sum of k numbers, only the final result is known to the parties; partial sums are not known to anybody. Other applications of our method include voting/rating over insecure channels and a rather elegant and efficient solution of the “two millionaires problem”. Then, while it is fairly obvious that a secure (bit) commitment between two parties is impossible without a one-way function, we show that it is possible if the number of parties is at least 3. We also show how our unconditionally secure (bit) commitment scheme for three parties can be used to arrange an unconditionally secure (bit) commitment between just two parties if they use a “dummy” (e.g., a computer) as the third party. We explain how our concept of a “dummy” is different from the well-known concept of a “trusted third party”. Based on a similar idea, we also offer an unconditionally secure k-n oblivious transfer protocol between two parties who use a “dummy”. We also suggest a protocol, without using a one-way function, for the so-called “mental poker”, i.e., a fair card dealing (and playing) over distance. Finally, we propose a secret sharing scheme where an advantage over Shamir's and other known secret sharing schemes is that nobody, including the dealer, ends up knowing the shares (of the secret) owned by any particular player. It should be mentioned that computational cost of our protocols is negligible to the point that all of them can be executed without a computer.
{"title":"Secrecy without one-way functions","authors":"D. Grigoriev, V. Shpilrain","doi":"10.1515/gcc-2013-0002","DOIUrl":"https://doi.org/10.1515/gcc-2013-0002","url":null,"abstract":"Abstract. We show that some problems in information security can be solved without using one-way functions. The latter are usually regarded as a central concept of cryptography, but the very existence of one-way functions depends on difficult conjectures in complexity theory, most notably on the notorious “” conjecture. This is why cryptographic primitives that do not employ one-way functions are often called “unconditionally secure”. In this paper, we suggest protocols for secure computation of the sum, product, and some other functions of two or more elements of an arbitrary constructible ring, without using any one-way functions. A new input that we offer here is that, in contrast with other proposals, we conceal “intermediate results” of a computation. For example, when we compute the sum of k numbers, only the final result is known to the parties; partial sums are not known to anybody. Other applications of our method include voting/rating over insecure channels and a rather elegant and efficient solution of the “two millionaires problem”. Then, while it is fairly obvious that a secure (bit) commitment between two parties is impossible without a one-way function, we show that it is possible if the number of parties is at least 3. We also show how our unconditionally secure (bit) commitment scheme for three parties can be used to arrange an unconditionally secure (bit) commitment between just two parties if they use a “dummy” (e.g., a computer) as the third party. We explain how our concept of a “dummy” is different from the well-known concept of a “trusted third party”. Based on a similar idea, we also offer an unconditionally secure k-n oblivious transfer protocol between two parties who use a “dummy”. We also suggest a protocol, without using a one-way function, for the so-called “mental poker”, i.e., a fair card dealing (and playing) over distance. Finally, we propose a secret sharing scheme where an advantage over Shamir's and other known secret sharing schemes is that nobody, including the dealer, ends up knowing the shares (of the secret) owned by any particular player. It should be mentioned that computational cost of our protocols is negligible to the point that all of them can be executed without a computer.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126459444","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. Results on algorithmic problems on strings that are given in a compressed form via straight-line programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.
{"title":"Algorithmics on SLP-compressed strings: A survey","authors":"Markus Lohrey","doi":"10.1515/gcc-2012-0016","DOIUrl":"https://doi.org/10.1515/gcc-2012-0016","url":null,"abstract":"Abstract. Results on algorithmic problems on strings that are given in a compressed form via straight-line programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133880804","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}
{"title":"An asymmetric generalisation of Artin monoids","authors":"D. Krammer","doi":"10.1515/gcc-2013-0010","DOIUrl":"https://doi.org/10.1515/gcc-2013-0010","url":null,"abstract":"Abstract. We propose a slight weakening of the definitions of Artin monoids and Coxeter monoids. We study one `infinite series' in detail.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122405938","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：