Groups Complexity Cryptology最新文献_第5页

Cryptanalysis of a combinatorial public key cryptosystem 组合公钥密码系统的密码分析

Groups Complexity Cryptology Pub Date : 2017-01-17 DOI: 10.1515/gcc-2017-0013

V. Roman’kov

引用次数: 14

A remark on spherical equations in free metabelian groups 自由亚系群中球方程的一个注释

Groups Complexity Cryptology Pub Date : 2017-01-17 DOI: 10.1515/gcc-2017-0012

E. Timoshenko

引用次数: 1

Generic hardness of the Boolean satisfiability problem 布尔可满足问题的一般硬度

Groups Complexity Cryptology Pub Date : 2017-01-12 DOI: 10.1515/gcc-2017-0008

A. Rybalov

引用次数: 2

The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5 长度不超过5的无扭转幂零群的同构问题

Groups Complexity Cryptology Pub Date : 2017-01-01 DOI: 10.1515/gcc-2017-0004

B. Eick, Ann-Kristin Engel

引用次数: 2

Pseudo-free families of finite computational elementary abelian p-groups 有限计算初等阿贝尔p群的无伪族

Groups Complexity Cryptology Pub Date : 2017-01-01 DOI: 10.1515/gcc-2017-0001

M. Anokhin

引用次数: 4

Log-space conjugacy problem in the Grigorchuk group Grigorchuk群的对数空间共轭问题

Groups Complexity Cryptology Pub Date : 2017-01-01 DOI: 10.1515/gcc-2017-0005

A. Myasnikov, S. Vassileva

引用次数: 5

Random nilpotent groups, polycyclic presentations, and Diophantine problems 随机幂零群，多环表示和丢番图问题

Groups Complexity Cryptology Pub Date : 2016-12-08 DOI: 10.1515/gcc-2017-0007

A. Garreta, A. Myasnikov, D. Ovchinnikov

{"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}

引用次数: 13

Conjugacy search problem and the Andrews–Curtis conjecture 共轭搜索问题与Andrews-Curtis猜想

Groups Complexity Cryptology Pub Date : 2016-09-01 DOI: 10.1515/gcc-2019-2005

Dmitry Panteleev, A. Ushakov

{"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}

引用次数: 6

The status of polycyclic group-based cryptography: A survey and open problems 基于多环群的密码学的现状:综述和开放问题

Groups Complexity Cryptology Pub Date : 2016-07-20 DOI: 10.1515/gcc-2016-0013

Jonathan Gryak, Delaram Kahrobaei

{"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}

引用次数: 26

Knapsack problem for nilpotent groups 幂零群的背包问题

Groups Complexity Cryptology Pub Date : 2016-06-28 DOI: 10.1515/gcc-2017-0006

A. Mishchenko, A. Treier

引用次数: 21