Abstract We discuss pitfalls in the security of the combinatorial public key cryptosystem based on Nielsen transformations inspired by the ElGamal cryptosystem proposed by Fine, Moldenhauer and Rosenberger. We introduce three different types of attacks to possible combinatorial public key encryption schemes and apply these attacks to the scheme corresponding to the cryptosystem under discussion. As a result of our observation, we show that under some natural assumptions the scheme is vulnerable to at least one of the proposed attacks.
{"title":"Cryptanalysis of a combinatorial public key cryptosystem","authors":"V. Roman’kov","doi":"10.1515/gcc-2017-0013","DOIUrl":"https://doi.org/10.1515/gcc-2017-0013","url":null,"abstract":"Abstract We discuss pitfalls in the security of the combinatorial public key cryptosystem based on Nielsen transformations inspired by the ElGamal cryptosystem proposed by Fine, Moldenhauer and Rosenberger. We introduce three different types of attacks to possible combinatorial public key encryption schemes and apply these attacks to the scheme corresponding to the cryptosystem under discussion. As a result of our observation, we show that under some natural assumptions the scheme is vulnerable to at least one of the proposed attacks.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"39 1","pages":"125 - 135"},"PeriodicalIF":0.0,"publicationDate":"2017-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80848723","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 I. Lysenok and A. Ushakov proved that the Diophantine problem for spherical quadric equations in free metabelian groups is solvable. The present paper proves this result by using the Magnus embedding.
I. Lysenok和A. Ushakov证明了自由亚谢群中球面二次方程的Diophantine问题是可解的。本文利用Magnus嵌入证明了这一结果。
{"title":"A remark on spherical equations in free metabelian groups","authors":"E. Timoshenko","doi":"10.1515/gcc-2017-0012","DOIUrl":"https://doi.org/10.1515/gcc-2017-0012","url":null,"abstract":"Abstract I. Lysenok and A. Ushakov proved that the Diophantine problem for spherical quadric equations in free metabelian groups is solvable. The present paper proves this result by using the Magnus embedding.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"2018 1","pages":"155 - 158"},"PeriodicalIF":0.0,"publicationDate":"2017-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87799832","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 follows from the famous result of Cook about the NP-completeness of the Boolean satisfiability problem that there is no polynomial algorithm for this problem if P ≠ N P {Pneq NP} . In this paper, we prove that the Boolean satisfiability problem remains computationally hard on polynomial strongly generic subsets of formulas provided P ≠ N P {Pneq NP} and P = B P P {P=BPP} . Boolean formulas are represented in the natural way by labeled binary trees.
{"title":"Generic hardness of the Boolean satisfiability problem","authors":"A. Rybalov","doi":"10.1515/gcc-2017-0008","DOIUrl":"https://doi.org/10.1515/gcc-2017-0008","url":null,"abstract":"Abstract It follows from the famous result of Cook about the NP-completeness of the Boolean satisfiability problem that there is no polynomial algorithm for this problem if P ≠ N P {Pneq NP} . In this paper, we prove that the Boolean satisfiability problem remains computationally hard on polynomial strongly generic subsets of formulas provided P ≠ N P {Pneq NP} and P = B P P {P=BPP} . Boolean formulas are represented in the natural way by labeled binary trees.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"9 1","pages":"151 - 154"},"PeriodicalIF":0.0,"publicationDate":"2017-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78493494","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 isomorphism problem for the finitely generated torsion free nilpotent groups of Hirsch length at most five. We show how this problem translates to solving an explicitly given set of polynomial equations. Based on this, we introduce a canonical form for each isomorphism type of finitely generated torsion free nilpotent group of Hirsch length at most 5 and, using a variation of our methods, we give an explicit description of its automorphisms.
{"title":"The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5","authors":"B. Eick, Ann-Kristin Engel","doi":"10.1515/gcc-2017-0004","DOIUrl":"https://doi.org/10.1515/gcc-2017-0004","url":null,"abstract":"Abstract We consider the isomorphism problem for the finitely generated torsion free nilpotent groups of Hirsch length at most five. We show how this problem translates to solving an explicitly given set of polynomial equations. Based on this, we introduce a canonical form for each isomorphism type of finitely generated torsion free nilpotent group of Hirsch length at most 5 and, using a variation of our methods, we give an explicit description of its automorphisms.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"9 1","pages":"55 - 75"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72769953","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 initiate the study of (weakly) pseudo-free families of computational elementary abelian p-groups, where p is an arbitrary fixed prime. We restrict ourselves to families of computational elementary abelian p-groups G d {G_{d}} such that for every index d, each element of G d {G_{d}} is represented by a single bit string of length polynomial in the length of d. First, we prove that pseudo-freeness and weak pseudo-freeness for families of computational elementary abelian p-groups are equivalent. Second, we give some necessary and sufficient conditions for a family of computational elementary abelian p-groups to be pseudo-free (provided that at least one of two additional conditions holds). Third, we establish some necessary and sufficient conditions for the existence of pseudo-free families of computational elementary abelian p-groups.
研究了计算初等阿贝尔p群的(弱)伪自由族，其中p是任意固定素数。我们将自己限制在计算初等阿贝尔p群G d {G_{d}}的族上，使得对于每一个指标d, G d {G_{d}}的每一个元素都由d的长度为多项式的单比特串表示。首先，我们证明了计算初等阿贝尔p群族的伪自由和弱伪自由是等价的。其次，我们给出了一类计算初等阿贝尔p群是伪自由的一些充分必要条件(前提是两个附加条件中至少有一个成立)。第三，建立了计算初等阿贝尔p群的无伪族存在的充分必要条件。
{"title":"Pseudo-free families of finite computational elementary abelian p-groups","authors":"M. Anokhin","doi":"10.1515/gcc-2017-0001","DOIUrl":"https://doi.org/10.1515/gcc-2017-0001","url":null,"abstract":"Abstract We initiate the study of (weakly) pseudo-free families of computational elementary abelian p-groups, where p is an arbitrary fixed prime. We restrict ourselves to families of computational elementary abelian p-groups G d {G_{d}} such that for every index d, each element of G d {G_{d}} is represented by a single bit string of length polynomial in the length of d. First, we prove that pseudo-freeness and weak pseudo-freeness for families of computational elementary abelian p-groups are equivalent. Second, we give some necessary and sufficient conditions for a family of computational elementary abelian p-groups to be pseudo-free (provided that at least one of two additional conditions holds). Third, we establish some necessary and sufficient conditions for the existence of pseudo-free families of computational elementary abelian p-groups.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"78 1","pages":"1 - 18"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88372391","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 prove that the conjugacy problem in the Grigorchuk group Γ has log-space complexity.
摘要本文证明了Grigorchuk群Γ中的共轭问题具有对数空间复杂度。
{"title":"Log-space conjugacy problem in the Grigorchuk group","authors":"A. Myasnikov, S. Vassileva","doi":"10.1515/gcc-2017-0005","DOIUrl":"https://doi.org/10.1515/gcc-2017-0005","url":null,"abstract":"Abstract In this paper we prove that the conjugacy problem in the Grigorchuk group Γ has log-space complexity.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":"77 - 85"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90111369","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 a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ 2 {tau_{2}} -groups). To do so, we show that these are precisely the groups with presentation of the form 〈 A , C ∣ [ a i , a j ] = ∏ t = 1 m c t λ t , i , j ( 1 ≤ i < j ≤ n ) , [ A , C ] = [ C , C ] = 1 〉 {langle A,Cmid[a_{i},a_{j}]=prod_{t=1}^{m}c_{t}^{lambda_{t,i,j}}(1leq i
{"title":"Random nilpotent groups, polycyclic presentations, and Diophantine problems","authors":"A. Garreta, A. Myasnikov, D. Ovchinnikov","doi":"10.1515/gcc-2017-0007","DOIUrl":"https://doi.org/10.1515/gcc-2017-0007","url":null,"abstract":"Abstract We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ 2 {tau_{2}} -groups). To do so, we show that these are precisely the groups with presentation of the form 〈 A , C ∣ [ a i , a j ] = ∏ t = 1 m c t λ t , i , j ( 1 ≤ i < j ≤ n ) , [ A , C ] = [ C , C ] = 1 〉 {langle A,Cmid[a_{i},a_{j}]=prod_{t=1}^{m}c_{t}^{lambda_{t,i,j}}(1leq i<j% leq n),,[A,C]=[C,C]=1rangle} , where A = { a 1 , … , a n } {A={a_{1},dots,a_{n}}} and C = { c 1 , … , c m } {C={c_{1},dots,c_{m}}} . Hence, a random G can be selected by fixing A and C, and then randomly choosing integers λ t , i , j {lambda_{t,i,j}} , with | λ t , i , j | ≤ ℓ {|lambda_{t,i,j}|leqell} for some ℓ {ell} . We prove that if m ≥ n - 1 ≥ 1 {mgeq n-1geq 1} , then the following hold asymptotically almost surely as ℓ → ∞ {elltoinfty} : the ring ℤ {mathbb{Z}} is e-definable in G, the Diophantine problem over G is undecidable, the maximal ring of scalars of G is ℤ {mathbb{Z}} , G is indecomposable as a direct product of non-abelian groups, and Z ( G ) = 〈 C 〉 {Z(G)=langle Crangle} . We further study when Z ( G ) ≤ Is ( G ′ ) {Z(G)leqoperatorname{Is}(G^{prime})} . Finally, we introduce similar models of random polycyclic groups and random f.g. nilpotent groups of any nilpotency step, possibly with torsion. We quickly see, however, that the latter yields finite groups a.a.s.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"129 1","pages":"115 - 99"},"PeriodicalIF":0.0,"publicationDate":"2016-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77257657","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 develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples AK ( n ) {operatorname{AK}(n)} . We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK ( 3 ) {operatorname{AK}(3)} . That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure. To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. We prove that automorphism moves can be applied to Akbulut–Kurby presentations. The improved technique allows us to enumerate balanced presentations AC-equivalent to AK ( 3 ) {operatorname{AK}(3)} with relations of lengths up to 20 (previous record was 17).
{"title":"Conjugacy search problem and the Andrews–Curtis conjecture","authors":"Dmitry Panteleev, A. Ushakov","doi":"10.1515/gcc-2019-2005","DOIUrl":"https://doi.org/10.1515/gcc-2019-2005","url":null,"abstract":"Abstract We develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples AK ( n ) {operatorname{AK}(n)} . We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK ( 3 ) {operatorname{AK}(3)} . That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure. To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. We prove that automorphism moves can be applied to Akbulut–Kurby presentations. The improved technique allows us to enumerate balanced presentations AC-equivalent to AK ( 3 ) {operatorname{AK}(3)} with relations of lengths up to 20 (previous record was 17).","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"20 1","pages":"43 - 60"},"PeriodicalIF":0.0,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/gcc-2019-2005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72537561","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 Polycyclic groups are natural generalizations of cyclic groups but with more complicated algorithmic properties. They are finitely presented and the word, conjugacy, and isomorphism decision problems are all solvable in these groups. Moreover, the non-virtually nilpotent ones exhibit an exponential growth rate. These properties make them suitable for use in group-based cryptography, which was proposed in 2004 by Eick and Kahrobaei [10]. Since then, many cryptosystems have been created that employ polycyclic groups. These include key exchanges such as non-commutative ElGamal, authentication schemes based on the twisted conjugacy problem, and secret sharing via the word problem. In response, heuristic and deterministic methods of cryptanalysis have been developed, including the length-based and linear decomposition attacks. Despite these efforts, there are classes of infinite polycyclic groups that remain suitable for cryptography. The analysis of algorithms for search and decision problems in polycyclic groups has also been developed. In addition to results for the aforementioned problems we present those concerning polycyclic representations, group morphisms, and orbit decidability. Though much progress has been made, many algorithmic and complexity problems remain unsolved; we conclude with a number of them. Of particular interest is to show that cryptosystems using infinite polycyclic groups are resistant to cryptanalysis on a quantum computer.
{"title":"The status of polycyclic group-based cryptography: A survey and open problems","authors":"Jonathan Gryak, Delaram Kahrobaei","doi":"10.1515/gcc-2016-0013","DOIUrl":"https://doi.org/10.1515/gcc-2016-0013","url":null,"abstract":"Abstract Polycyclic groups are natural generalizations of cyclic groups but with more complicated algorithmic properties. They are finitely presented and the word, conjugacy, and isomorphism decision problems are all solvable in these groups. Moreover, the non-virtually nilpotent ones exhibit an exponential growth rate. These properties make them suitable for use in group-based cryptography, which was proposed in 2004 by Eick and Kahrobaei [10]. Since then, many cryptosystems have been created that employ polycyclic groups. These include key exchanges such as non-commutative ElGamal, authentication schemes based on the twisted conjugacy problem, and secret sharing via the word problem. In response, heuristic and deterministic methods of cryptanalysis have been developed, including the length-based and linear decomposition attacks. Despite these efforts, there are classes of infinite polycyclic groups that remain suitable for cryptography. The analysis of algorithms for search and decision problems in polycyclic groups has also been developed. In addition to results for the aforementioned problems we present those concerning polycyclic representations, group morphisms, and orbit decidability. Though much progress has been made, many algorithmic and complexity problems remain unsolved; we conclude with a number of them. Of particular interest is to show that cryptosystems using infinite polycyclic groups are resistant to cryptanalysis on a quantum computer.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"5 1","pages":"171 - 186"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72938907","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 work we investigate the group version of the well known knapsack problem in the class of nilpotent groups. The main result of this paper is that the knapsack problem is undecidable for any torsion-free group of nilpotency class 2 if the rank of the derived subgroup is at least 316. Also, we extend our result to certain classes of polycyclic groups, linear groups, and nilpotent groups of nilpotency class greater than or equal to 2.
{"title":"Knapsack problem for nilpotent groups","authors":"A. Mishchenko, A. Treier","doi":"10.1515/gcc-2017-0006","DOIUrl":"https://doi.org/10.1515/gcc-2017-0006","url":null,"abstract":"Abstract In this work we investigate the group version of the well known knapsack problem in the class of nilpotent groups. The main result of this paper is that the knapsack problem is undecidable for any torsion-free group of nilpotency class 2 if the rank of the derived subgroup is at least 316. Also, we extend our result to certain classes of polycyclic groups, linear groups, and nilpotent groups of nilpotency class greater than or equal to 2.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"20 1","pages":"87 - 98"},"PeriodicalIF":0.0,"publicationDate":"2016-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81777061","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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