Abstract In this paper we study the asymptotic probability that a random equation in a finitely generated free group F is solvable in F. For one-variable equations this probability is zero, but for split equations, i.e., equations of the form v(x 1, . . . , xk ) = g, g ∈ F, the probability is strictly between zero and one if k ≥ rank(F) ≥ 2. As a consequence the endomorphism problem in F has intermediate asymptotic density, and we obtain the first natural algebraic examples of subsets of intermediate density in free groups of rank larger than two.
{"title":"Random equations in free groups","authors":"R. Gilman, A. Myasnikov, V. Roman’kov","doi":"10.1515/gcc.2011.010","DOIUrl":"https://doi.org/10.1515/gcc.2011.010","url":null,"abstract":"Abstract In this paper we study the asymptotic probability that a random equation in a finitely generated free group F is solvable in F. For one-variable equations this probability is zero, but for split equations, i.e., equations of the form v(x 1, . . . , xk ) = g, g ∈ F, the probability is strictly between zero and one if k ≥ rank(F) ≥ 2. As a consequence the endomorphism problem in F has intermediate asymptotic density, and we obtain the first natural algebraic examples of subsets of intermediate density in free groups of rank larger than two.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"27 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":"125253857","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 present a new algorithm to find apartments in coset geometries. It permits to obtain some missing results in a paper by Buekenhout and Leemans. We compare the performance of our algorithm to the one given by Buekenhout and Leemans.
{"title":"A new algorithm to find apartments in coset geometries","authors":"Thomas Connor, D. Leemans","doi":"10.1515/gcc-2013-0006","DOIUrl":"https://doi.org/10.1515/gcc-2013-0006","url":null,"abstract":"Abstract. We present a new algorithm to find apartments in coset geometries. It permits to obtain some missing results in a paper by Buekenhout and Leemans. We compare the performance of our algorithm to the one given by Buekenhout and Leemans.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"95 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":"130346492","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}
G. Baumslag, Yegor Bryukhov, B. Fine, Douglas R. Troeger
Abstract Challenge response methods are increasingly used to enhance password security. In this paper we present a very secure method for challenge response password verification using combinatorial group theory. This method, which relies on the group randomizer system, a subset of the MAGNUS computer algebra system, handles most of the present problems with challenge response systems. Theoretical security is based on several results in asymptotic group theory.
{"title":"Challenge response password security using combinatorial group theory","authors":"G. Baumslag, Yegor Bryukhov, B. Fine, Douglas R. Troeger","doi":"10.1515/gcc.2010.005","DOIUrl":"https://doi.org/10.1515/gcc.2010.005","url":null,"abstract":"Abstract Challenge response methods are increasingly used to enhance password security. In this paper we present a very secure method for challenge response password verification using combinatorial group theory. This method, which relies on the group randomizer system, a subset of the MAGNUS computer algebra system, handles most of the present problems with challenge response systems. Theoretical security is based on several results in asymptotic group theory.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"45 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131578672","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 is the first in a two-part survey of current techniques in algebraic cryptanalysis. After introducing the basic setup of algebraic attacks and discussing several attack scenarios for symmetric cryptosystems, public key cryptosystems, and stream ciphers, we discuss a number of individual methods. The XL, XSL, and MutantXL attacks are based on linearization techniques for multivariate polynomial systems. Then we look at Gröbner basis and border bases methods. In the last section we introduce attacks based on integer programming techniques and try them in some concrete cases.
{"title":"Algebraic Attacks Galore!","authors":"M. Kreuzer","doi":"10.1515/GCC.2009.231","DOIUrl":"https://doi.org/10.1515/GCC.2009.231","url":null,"abstract":"This is the first in a two-part survey of current techniques in algebraic cryptanalysis. After introducing the basic setup of algebraic attacks and discussing several attack scenarios for symmetric cryptosystems, public key cryptosystems, and stream ciphers, we discuss a number of individual methods. The XL, XSL, and MutantXL attacks are based on linearization techniques for multivariate polynomial systems. Then we look at Gröbner basis and border bases methods. In the last section we introduce attacks based on integer programming techniques and try them in some concrete cases.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"35 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":"124298190","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 For many groups the structure of finitely generated subgroups is generically simple. That is with asymptotic density equal to one a randomly chosen finitely generated subgroup has a particular well-known and easily analyzed structure. For example a result of D. B. A. Epstein says that a finitely generated subgroup of GL(n, ℝ) is generically a free group. We say that a group G has the generic free group property if any finitely generated subgroup is generically a free group. Further G has the strong generic free group property if given randomly chosen elements g 1, . . . , gn in G then generically they are a free basis for the free subgroup they generate. In this paper we show that for any arbitrary free product of finitely generated infinite groups satisfies the strong generic free group property. There are also extensions to more general amalgams - free products with amalgamation and HNN groups. These results have implications in cryptography. In particular several cryptosystems use random choices of subgroups as hard cryptographic problems. In groups with the generic free group property any such cryptosystem may be attackable by a length based attack.
摘要对于许多群，有限生成子群的结构一般是简单的。也就是说，当密度渐近等于1时，随机选择的有限生成的子群具有特定的众所周知的易于分析的结构。例如，D. B. a . Epstein的一个结果说，GL(n, l)的有限生成子群一般是一个自由群。如果任何有限生成的子群是一般自由群，则群G具有一般自由群的性质。若给定随机选取的元素g1，…，则G具有强一般自由群性质。， gn在G中，那么一般来说它们是它们生成的自由子群的自由基。本文证明了有限生成无限群的任意自由积满足强一般自由群的性质。也有扩展到更一般的无汞合金产品与合并和HNN组。这些结果对密码学有影响。特别是一些密码系统使用子群的随机选择作为硬密码问题。在具有一般自由群属性的群中，任何这样的密码系统都可以被基于长度的攻击攻击。
{"title":"Generic Subgroups of Group Amalgams","authors":"B. Fine, A. Myasnikov, G. Rosenberger","doi":"10.1515/GCC.2009.51","DOIUrl":"https://doi.org/10.1515/GCC.2009.51","url":null,"abstract":"Abstract For many groups the structure of finitely generated subgroups is generically simple. That is with asymptotic density equal to one a randomly chosen finitely generated subgroup has a particular well-known and easily analyzed structure. For example a result of D. B. A. Epstein says that a finitely generated subgroup of GL(n, ℝ) is generically a free group. We say that a group G has the generic free group property if any finitely generated subgroup is generically a free group. Further G has the strong generic free group property if given randomly chosen elements g 1, . . . , gn in G then generically they are a free basis for the free subgroup they generate. In this paper we show that for any arbitrary free product of finitely generated infinite groups satisfies the strong generic free group property. There are also extensions to more general amalgams - free products with amalgamation and HNN groups. These results have implications in cryptography. In particular several cryptosystems use random choices of subgroups as hard cryptographic problems. In groups with the generic free group property any such cryptosystem may be attackable by a length based attack.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"245 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":"121982697","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 introduce the standard computable-model-theoretic concepts of a computable group and a computable field, and use them to illustrate the sorts of questions about groups and fields which computability theorists investigate. This article is intended for group theorists with some background in algorithmic questions, such as the undecidability of the word problem and the conjugacy problem for finitely presented groups.
{"title":"An introduction to computable model theory on groups and fields","authors":"Russell G. Miller","doi":"10.1515/gcc.2011.002","DOIUrl":"https://doi.org/10.1515/gcc.2011.002","url":null,"abstract":"Abstract We introduce the standard computable-model-theoretic concepts of a computable group and a computable field, and use them to illustrate the sorts of questions about groups and fields which computability theorists investigate. This article is intended for group theorists with some background in algorithmic questions, such as the undecidability of the word problem and the conjugacy problem for finitely presented groups.","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":"114634436","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 argue that it is unnatural and undesirable to use the non-uniform model of complexity for practice-oriented security reductions in cryptography.
摘要我们认为，在密码学中使用非统一的复杂性模型来降低面向实践的安全性是不自然和不可取的。
{"title":"Another look at non-uniformity","authors":"N. Koblitz, A. Menezes","doi":"10.1515/gcc-2013-0008","DOIUrl":"https://doi.org/10.1515/gcc-2013-0008","url":null,"abstract":"Abstract. We argue that it is unnatural and undesirable to use the non-uniform model of complexity for practice-oriented security reductions in cryptography.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"26 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":"117048613","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 the multiple conjugacy search problem over a subclass of partially commutative groups and experimentally attack it with a genetic algorithm hybridised with a “length attack”. We detail symbolic computation of words over the groups, constructing functions which measure certain statistics of those words. By experimentation, the hybrid algorithm is shown to be effective, showing that the standard conjugacy search problem is harder than the multiple conjugacy search problem for our groups. Moreover, some intuitive methods of increasing problem difficulty are overcome by the algorithm, and in fact make the problem easier to solve. We show our algorithm is efficient, comparing well with traditional approaches in groups that are statistically similar. Finally, via “approximation” of braid groups by our subclass, we consider implications of the attack on certain cryptosystems, pointing to further work in the discipline of group-theoretic cryptography.
{"title":"Evolutionary algorithm solution of the multiple conjugacy search problem in groups, and its applications to cryptography","authors":"M. J. Craven, H. C. Jimbo","doi":"10.1515/gcc-2012-0002","DOIUrl":"https://doi.org/10.1515/gcc-2012-0002","url":null,"abstract":"Abstract. We consider the multiple conjugacy search problem over a subclass of partially commutative groups and experimentally attack it with a genetic algorithm hybridised with a “length attack”. We detail symbolic computation of words over the groups, constructing functions which measure certain statistics of those words. By experimentation, the hybrid algorithm is shown to be effective, showing that the standard conjugacy search problem is harder than the multiple conjugacy search problem for our groups. Moreover, some intuitive methods of increasing problem difficulty are overcome by the algorithm, and in fact make the problem easier to solve. We show our algorithm is efficient, comparing well with traditional approaches in groups that are statistically similar. Finally, via “approximation” of braid groups by our subclass, we consider implications of the attack on certain cryptosystems, pointing to further work in the discipline of group-theoretic cryptography.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"40 7 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":"132761170","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 explain and perform the steps for an (n,t) secret sharing scheme based on the closest vector theorem. We then compare this scheme and its complexity to the secret sharing schemes of both Shamir and Panagopoulos. Finally we modify the (n,t) secret sharing scheme to a private key cryptosystem.
{"title":"A secret sharing scheme based on the Closest Vector Theorem and a modification to a private key cryptosystem","authors":"B. Fine, A. Moldenhauer, G. Rosenberger","doi":"10.1515/gcc-2013-0012","DOIUrl":"https://doi.org/10.1515/gcc-2013-0012","url":null,"abstract":"Abstract. We explain and perform the steps for an (n,t) secret sharing scheme based on the closest vector theorem. We then compare this scheme and its complexity to the secret sharing schemes of both Shamir and Panagopoulos. Finally we modify the (n,t) secret sharing scheme to a private key cryptosystem.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"32 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":"114487368","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 the problem of constructing continuous cryptographic primitives. We present several candidates for continuous hard-to-invert functions. To formulate these candidates, we introduce constructions based on tropical and supertropical circuits.
{"title":"Continuous hard-to-invert functions and biometric authentication","authors":"D. Grigoriev, S. Nikolenko","doi":"10.1515/gcc-2012-0004","DOIUrl":"https://doi.org/10.1515/gcc-2012-0004","url":null,"abstract":"Abstract. We consider the problem of constructing continuous cryptographic primitives. We present several candidates for continuous hard-to-invert functions. To formulate these candidates, we introduce constructions based on tropical and supertropical circuits.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"29 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":"117092423","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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