Abstract In this paper, we propose a more secure version of a Cayley hash function which is based on the linear functions. It is a practical parallelizable hash function.
摘要本文提出了一种基于线性函数的更安全的Cayley哈希函数。它是一个实用的可并行散列函数。
{"title":"More secure version of a Cayley hash function","authors":"Mohammad Hossein Ghaffari, Z. Mostaghim","doi":"10.1515/gcc-2018-0002","DOIUrl":"https://doi.org/10.1515/gcc-2018-0002","url":null,"abstract":"Abstract In this paper, we propose a more secure version of a Cayley hash function which is based on the linear functions. It is a practical parallelizable hash function.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"20 1","pages":"29 - 32"},"PeriodicalIF":0.0,"publicationDate":"2018-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83679346","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 word problem of a group G = 〈 Σ 〉 {G=langleSigmarangle} can be defined as the set of formal words in Σ * {Sigma^{*}} that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of ℤ 2 {mathbb{Z}^{2}} is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of ℤ n {mathbb{Z}^{n}} is a multiple context-free language for any n.
群G= < Σ > {G=langleSigmarangle}的词问题可以定义为Σ * {Sigma^{*}}中表示G中的恒等的形式词的集合。当被视为形式语言时，这给出了群的类与形式语言的类之间的强联系。例如，anj ? s ? mov证明了一个群是有限的当且仅当它的词问题是一种规则语言，Muller和Schupp证明了一个群是虚拟自由的当且仅当它的词问题是一种上下文自由语言。最近，Salvati证明了{mathbb{Z}^{2}}的词问题是一个多重上下文无关的语言，给出了一个多重上下文无关的自然词问题的第一个例子，但不是上下文无关的。我们推广了Salvati的结果，证明了对于任意n, n {mathbb{Z}^{n}}的词问题是一个多上下文无关的语言。
{"title":"The word problem of ℤ n is a multiple context-free language","authors":"M. Ho","doi":"10.1515/gcc-2018-0003","DOIUrl":"https://doi.org/10.1515/gcc-2018-0003","url":null,"abstract":"Abstract The word problem of a group G = 〈 Σ 〉 {G=langleSigmarangle} can be defined as the set of formal words in Σ * {Sigma^{*}} that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of ℤ 2 {mathbb{Z}^{2}} is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of ℤ n {mathbb{Z}^{n}} is a multiple context-free language for any n.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"108 1","pages":"15 - 9"},"PeriodicalIF":0.0,"publicationDate":"2018-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73057601","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 will offer a new symmetric-key cryptographic scheme which is based on the existence of exponentially distorted subgroups in arithmetic groups. Aside from this, we will also provide new examples of distorted subgroups in SL n ( ℤ [ x ] ) {mathrm{SL}_{n}(mathbb{Z}[x])} which can be utilized for the same purpose.
{"title":"Some applications of arithmetic groups in cryptography","authors":"Delaram Kahrobaei, Keivan Mallahi-Karai","doi":"10.1515/gcc-2019-2002","DOIUrl":"https://doi.org/10.1515/gcc-2019-2002","url":null,"abstract":"Abstract In this paper, we will offer a new symmetric-key cryptographic scheme which is based on the existence of exponentially distorted subgroups in arithmetic groups. Aside from this, we will also provide new examples of distorted subgroups in SL n ( ℤ [ x ] ) {mathrm{SL}_{n}(mathbb{Z}[x])} which can be utilized for the same purpose.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"15 1","pages":"25 - 33"},"PeriodicalIF":0.0,"publicationDate":"2018-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73268158","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}
Pub Date : 2017-10-15DOI: 10.46298/jgcc.2021.13.1.6521
L. Babinkostova, A. Hern'andez-Espiet, H. Kim
We generalize the notions of elliptic pseudoprimes and elliptic Carmichael numbers introduced by Silverman to analogues of Euler-Jacobi and strong pseudoprimes. We investigate the relationships among Euler Elliptic Carmichael numbers , strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I: The former two of these are introduced in this paper, and the latter two of these were introduced by Mazur (1973) and Silverman (2012) respectively. In particular, we expand upon a previous work of Babinkostova et al. by proving a conjecture about the density of certain elliptic Korselt numbers of Type I that are products of anomalous primes. Comment: Revised for publication. 33 pages
{"title":"On Types of Elliptic Pseudoprimes","authors":"L. Babinkostova, A. Hern'andez-Espiet, H. Kim","doi":"10.46298/jgcc.2021.13.1.6521","DOIUrl":"https://doi.org/10.46298/jgcc.2021.13.1.6521","url":null,"abstract":"We generalize the notions of elliptic pseudoprimes and elliptic Carmichael\u0000numbers introduced by Silverman to analogues of Euler-Jacobi and strong\u0000pseudoprimes. We investigate the relationships among Euler Elliptic Carmichael\u0000numbers , strong elliptic Carmichael numbers, products of anomalous primes and\u0000elliptic Korselt numbers of Type I: The former two of these are introduced in\u0000this paper, and the latter two of these were introduced by Mazur (1973) and\u0000Silverman (2012) respectively. In particular, we expand upon a previous work of\u0000Babinkostova et al. by proving a conjecture about the density of certain\u0000elliptic Korselt numbers of Type I that are products of anomalous primes.\u0000Comment: Revised for publication. 33 pages","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75202344","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 the class of groups with k-multiple context-free word problem is closed under graphs of groups with finite edge groups.
摘要我们证明了具有k个无上下文词问题的群在有限边群群的图下是封闭的。
{"title":"Closure properties in the class of multiple context-free groups","authors":"Robert P. Kropholler, Davide Spriano","doi":"10.1515/gcc-2019-2004","DOIUrl":"https://doi.org/10.1515/gcc-2019-2004","url":null,"abstract":"Abstract We show that the class of groups with k-multiple context-free word problem is closed under graphs of groups with finite edge groups.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"30 1","pages":"1 - 15"},"PeriodicalIF":0.0,"publicationDate":"2017-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73672381","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 Recent work by Grigoriev and Shpilrain [8] suggests looking at the tropical semiring for cryptographic schemes. In this contribution we explore the tropical analogue of the Hessian pencil of plane cubic curves as a source of group-based cryptography. Using elementary tropical geometry on the tropical Hessian curves, we derive the addition and doubling formulas induced from their Jacobian and investigate the discrete logarithm problem in this group. We show that the DLP is solvable when restricted to integral points on the tropical Hesse curve, and hence inadequate for cryptographic applications. Consideration of point duplication, however, provides instances of solvable chaotic maps producing random sequences and thus a source of fast keyed hash functions.
{"title":"Cryptography from the tropical Hessian pencil","authors":"J. Chauvet, É. Mahé","doi":"10.1515/gcc-2017-0002","DOIUrl":"https://doi.org/10.1515/gcc-2017-0002","url":null,"abstract":"Abstract Recent work by Grigoriev and Shpilrain [8] suggests looking at the tropical semiring for cryptographic schemes. In this contribution we explore the tropical analogue of the Hessian pencil of plane cubic curves as a source of group-based cryptography. Using elementary tropical geometry on the tropical Hessian curves, we derive the addition and doubling formulas induced from their Jacobian and investigate the discrete logarithm problem in this group. We show that the DLP is solvable when restricted to integral points on the tropical Hesse curve, and hence inadequate for cryptographic applications. Consideration of point duplication, however, provides instances of solvable chaotic maps producing random sequences and thus a source of fast keyed hash functions.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":"19 - 29"},"PeriodicalIF":0.0,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83634808","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 establish a lower bound on the spectral gap of the Laplace operator on special linear groups using conic optimisation. In particular, this provides a constructive (but computer assisted) proof that these groups have the Kazhdan property (T). Software for such optimisation for other finitely presented groups is provided.
{"title":"Certifying numerical estimates of spectral gaps","authors":"M. Kaluba, P. Nowak","doi":"10.1515/gcc-2018-0004","DOIUrl":"https://doi.org/10.1515/gcc-2018-0004","url":null,"abstract":"Abstract We establish a lower bound on the spectral gap of the Laplace operator on special linear groups using conic optimisation. In particular, this provides a constructive (but computer assisted) proof that these groups have the Kazhdan property (T). Software for such optimisation for other finitely presented groups is provided.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"18 1","pages":"33 - 41"},"PeriodicalIF":0.0,"publicationDate":"2017-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83321213","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 class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic. We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [6]. We give examples of abstract groups lying in this class, but also show that Gersten’s free by cyclic group does not. This implies that it has no faithful linear representation of any dimension over any field of positive characteristic, nor can it be embedded in any complex unitary group.
{"title":"Free by cyclic groups and linear groups with restricted unipotent elements","authors":"J. Button","doi":"10.1515/gcc-2017-0009","DOIUrl":"https://doi.org/10.1515/gcc-2017-0009","url":null,"abstract":"Abstract We introduce the class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic. We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [6]. We give examples of abstract groups lying in this class, but also show that Gersten’s free by cyclic group does not. This implies that it has no faithful linear representation of any dimension over any field of positive characteristic, nor can it be embedded in any complex unitary group.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":"137 - 149"},"PeriodicalIF":0.0,"publicationDate":"2017-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76712813","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 give an elementary proof of the group law for elliptic curves using explicit formulas.
摘要利用显式公式给出椭圆曲线群律的初等证明。
{"title":"An elementary proof of the group law for elliptic curves","authors":"Stefan Friedl","doi":"10.1515/gcc-2017-0010","DOIUrl":"https://doi.org/10.1515/gcc-2017-0010","url":null,"abstract":"Abstract We give an elementary proof of the group law for elliptic curves using explicit formulas.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"8 1","pages":"117 - 123"},"PeriodicalIF":0.0,"publicationDate":"2017-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89539567","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 study equationally Noetherian and 𝐪 ω {{mathbf{q}_{omega}}} -compact varieties of groups, rings and monoids. Moreover, we describe equationally Noetherian direct powers for these algebraic structures.
{"title":"Direct products, varieties, and compactness conditions","authors":"M. Shahryari, A. Shevlyakov","doi":"10.1515/gcc-2017-0011","DOIUrl":"https://doi.org/10.1515/gcc-2017-0011","url":null,"abstract":"Abstract We study equationally Noetherian and 𝐪 ω {{mathbf{q}_{omega}}} -compact varieties of groups, rings and monoids. Moreover, we describe equationally Noetherian direct powers for these algebraic structures.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"6 1","pages":"159 - 166"},"PeriodicalIF":0.0,"publicationDate":"2017-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91341413","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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