Pub Date : 2020-02-18DOI: 10.46298/jgcc.2020.12.2.6200
A. Bishop, Michal Ferov
Small cancellation groups form an interesting class with many desirable properties. It is a well-known fact that small cancellation groups are generic; however, all previously known results of their genericity are asymptotic and provide no information about "small" group presentations. In this note, we give closed-form formulas for both lower and upper bounds on the density of small cancellation presentations, and compare our results with experimental data. Comment: 18 pages, 12 figures
{"title":"Density of Metric Small Cancellation in Finitely Presented Groups","authors":"A. Bishop, Michal Ferov","doi":"10.46298/jgcc.2020.12.2.6200","DOIUrl":"https://doi.org/10.46298/jgcc.2020.12.2.6200","url":null,"abstract":"Small cancellation groups form an interesting class with many desirable\u0000properties. It is a well-known fact that small cancellation groups are generic;\u0000however, all previously known results of their genericity are asymptotic and\u0000provide no information about \"small\" group presentations. In this note, we give\u0000closed-form formulas for both lower and upper bounds on the density of small\u0000cancellation presentations, and compare our results with experimental data.\u0000Comment: 18 pages, 12 figures","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89077683","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 study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.
{"title":"On group automorphisms in universal algebraic geometry","authors":"A. Shevlyakov","doi":"10.1515/gcc-2019-2008","DOIUrl":"https://doi.org/10.1515/gcc-2019-2008","url":null,"abstract":"Abstract In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"35 1","pages":"115 - 121"},"PeriodicalIF":0.0,"publicationDate":"2019-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90224165","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, given a finitely generated group G as the coordinate group of a finite system of equations over a torsion-free hyperbolic group Γ, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from G to Γ as compositions of factorizations through Γ-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of Γ-limit groups as iterated generalized doubles over Γ.
{"title":"Effective construction of covers of canonical Hom-diagrams for equations over torsion-free hyperbolic groups","authors":"O. Kharlampovich, A. Myasnikov, Alexander Taam","doi":"10.1515/gcc-2019-2010","DOIUrl":"https://doi.org/10.1515/gcc-2019-2010","url":null,"abstract":"Abstract We show that, given a finitely generated group G as the coordinate group of a finite system of equations over a torsion-free hyperbolic group Γ, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from G to Γ as compositions of factorizations through Γ-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of Γ-limit groups as iterated generalized doubles over Γ.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"32 1","pages":"83 - 101"},"PeriodicalIF":0.0,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82935143","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 Secret sharing allows one to share a piece of information among n participants in a way that only qualified subsets of participants can recover the secret whereas others cannot. Some of these participants involved may, however, want to forge their shares of the secret(s) in order to cheat other participants. Various cheater identifiable techniques have been devised in order to identify such cheaters in secret sharing schemes. On the other hand, Ramp secret sharing schemes are a practically efficient variant of usual secret sharing schemes with reduced share size and some loss in security. Ramp secret sharing schemes have many applications in secure information storage, information-theoretic private information retrieval and secret image sharing due to producing relatively smaller shares. However, to the best of our knowledge, there does not exist any cheater identifiable ramp secret sharing scheme. In this paper we define the security model for cheater identifiable ramp secret sharing schemes and provide two constructions for cheater identifiable ramp secret sharing schemes. In addition, the second construction is secure against rushing cheaters who are allowed to submit their shares during secret reconstruction after observing other participants’ responses in one round. Also, we do not make any computational assumptions for the cheaters, i.e., cheaters may be equipped with unlimited time and resources, yet, the cheating probability would be bounded above by a very small positive number.
{"title":"Ramp secret sharing with cheater identification in presence of rushing cheaters","authors":"Jyotirmoy Pramanik, A. Adhikari","doi":"10.1515/gcc-2019-2006","DOIUrl":"https://doi.org/10.1515/gcc-2019-2006","url":null,"abstract":"Abstract Secret sharing allows one to share a piece of information among n participants in a way that only qualified subsets of participants can recover the secret whereas others cannot. Some of these participants involved may, however, want to forge their shares of the secret(s) in order to cheat other participants. Various cheater identifiable techniques have been devised in order to identify such cheaters in secret sharing schemes. On the other hand, Ramp secret sharing schemes are a practically efficient variant of usual secret sharing schemes with reduced share size and some loss in security. Ramp secret sharing schemes have many applications in secure information storage, information-theoretic private information retrieval and secret image sharing due to producing relatively smaller shares. However, to the best of our knowledge, there does not exist any cheater identifiable ramp secret sharing scheme. In this paper we define the security model for cheater identifiable ramp secret sharing schemes and provide two constructions for cheater identifiable ramp secret sharing schemes. In addition, the second construction is secure against rushing cheaters who are allowed to submit their shares during secret reconstruction after observing other participants’ responses in one round. Also, we do not make any computational assumptions for the cheaters, i.e., cheaters may be equipped with unlimited time and resources, yet, the cheating probability would be bounded above by a very small positive number.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"79 1","pages":"103 - 113"},"PeriodicalIF":0.0,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81800041","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}
M. R. M. Shamsabad, S. M. Dehnavi, A. M. Rishakani
Abstract MDS diffusion layers are critical components in the design of symmetric ciphers. In this paper, after introducing some new algebraic structures, we provide new MDS matrices over special types of R-modules. With the help of the proposed methodology, we have more flexibility in designing software-oriented diffusion layers. Most notably, we construct randomized and/or nonlinear MDS diffusion layers, based upon the presented theoretical results, and discuss the resistance of the presented diffusion layers against various kinds of cryptanalysis, compared with classical linear diffusion layers.
{"title":"Randomized nonlinear software-oriented MDS diffusion layers","authors":"M. R. M. Shamsabad, S. M. Dehnavi, A. M. Rishakani","doi":"10.1515/gcc-2019-2011","DOIUrl":"https://doi.org/10.1515/gcc-2019-2011","url":null,"abstract":"Abstract MDS diffusion layers are critical components in the design of symmetric ciphers. In this paper, after introducing some new algebraic structures, we provide new MDS matrices over special types of R-modules. With the help of the proposed methodology, we have more flexibility in designing software-oriented diffusion layers. Most notably, we construct randomized and/or nonlinear MDS diffusion layers, based upon the presented theoretical results, and discuss the resistance of the presented diffusion layers against various kinds of cryptanalysis, compared with classical linear diffusion layers.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"15 1","pages":"123 - 131"},"PeriodicalIF":0.0,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80403736","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":"Constructing a pseudo-free family of finite computational groups under the general integer factoring intractability assumption","authors":"M. Anokhin","doi":"10.1515/gcc-2019-2009","DOIUrl":"https://doi.org/10.1515/gcc-2019-2009","url":null,"abstract":"Abstract We provide a correct version of Remark 3.5 of the paper mentioned in the title. Also, we fix a typo in Remark 4.4 of that paper.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"11 1","pages":"133 - 134"},"PeriodicalIF":0.0,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75500031","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 : 2019-05-29DOI: 10.46298/jgcc.2020.12.1.6059
Jordi Delgado, Pedro V. Silva
A $vee$-complement of a subgroup $H leqslant mathbb{F}_n$ is a subgroup $K leqslant mathbb{F}_n$ such that $H vee K = mathbb{F}_n$. If we also ask $K$ to have trivial intersection with $H$, then we say that $K$ is a $oplus$-complement of $H$. The minimum possible rank of a $vee$-complement (resp. $oplus$-complement) of $H$ is called the $vee$-corank (resp. $oplus$-corank) of $H$. We use Stallings automata to study these notions and the relations between them. In particular, we characterize when complements exist, compute the $vee$-corank, and provide language-theoretical descriptions of the sets of cyclic complements. Finally, we prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds. Comment: 27 pages, 5 figures
子组$H leqslant mathbb{F}_n$的$vee$ -补码是子组$Kleqslant mathbb{F}_n$，这样$H vee K = mathbb{F}_n$。如果我们还要求$K$与$H$有平凡的交点，那么我们说$K$是$H$的$oplus$ -补。$vee$ -补码的最小可能秩。$H$的$oplus$ -补充)称为$vee$ -corank(参见:$oplus$ -corank)的$H$。我们使用斯托林斯自动机来研究这些概念以及它们之间的关系。特别地，我们描述了何时是互补的，计算了$vee$ - ank，并提供了循环互补集的语言理论描述。最后，我们证明了两个秩的概念在两种循环补的子群上重合。点评:27页，5张图
{"title":"On the lattice of subgroups of a free group: complements and rank","authors":"Jordi Delgado, Pedro V. Silva","doi":"10.46298/jgcc.2020.12.1.6059","DOIUrl":"https://doi.org/10.46298/jgcc.2020.12.1.6059","url":null,"abstract":"A $vee$-complement of a subgroup $H leqslant mathbb{F}_n$ is a subgroup $K\u0000leqslant mathbb{F}_n$ such that $H vee K = mathbb{F}_n$. If we also ask $K$\u0000to have trivial intersection with $H$, then we say that $K$ is a\u0000$oplus$-complement of $H$. The minimum possible rank of a $vee$-complement\u0000(resp. $oplus$-complement) of $H$ is called the $vee$-corank (resp.\u0000$oplus$-corank) of $H$. We use Stallings automata to study these notions and\u0000the relations between them. In particular, we characterize when complements\u0000exist, compute the $vee$-corank, and provide language-theoretical descriptions\u0000of the sets of cyclic complements. Finally, we prove that the two notions of\u0000corank coincide on subgroups that admit cyclic complements of both kinds.\u0000Comment: 27 pages, 5 figures","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84682301","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 suggest several automaton groups as platforms for Anshel–Anshel–Goldfeld key agreement metascheme. They include Grigorchuk and universal Grigorchuk groups, Hanoi 3-towers group, the Basilica group and a subgroup of the affine group Aff4(ℤ){mathrm{Aff}_{4}(mathbb{Z})}.
{"title":"Key agreement based on automaton groups","authors":"R. Grigorchuk, D. Grigoriev","doi":"10.1515/gcc-2019-2012","DOIUrl":"https://doi.org/10.1515/gcc-2019-2012","url":null,"abstract":"Abstract We suggest several automaton groups as platforms for Anshel–Anshel–Goldfeld key agreement metascheme. They include Grigorchuk and universal Grigorchuk groups, Hanoi 3-towers group, the Basilica group and a subgroup of the affine group Aff4(ℤ){mathrm{Aff}_{4}(mathbb{Z})}.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"100 1","pages":"77 - 81"},"PeriodicalIF":0.0,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80023970","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 An improved version of the Anshel–Anshel–Goldfeld (AAG) algebraic cryptographic key-exchange scheme, that is in particular resistant against the Tsaban linear span cryptanalysis, is established. Unlike the original version, that is based on the intractability of the simultaneous conjugacy search problem for the platform group, the proposed version is based on harder simultaneous membership-conjugacy search problems, and the membership problem needs to be solved for a subset of the platform group that can be easily and efficiently built to be very complicated and without any good structure. A number of other hard problems need to be solved first before start solving the simultaneous membership-conjugacy search problem to obtain the exchanged key.
{"title":"An improved version of the AAG cryptographic protocol","authors":"V. Roman’kov","doi":"10.1515/gcc-2019-2003","DOIUrl":"https://doi.org/10.1515/gcc-2019-2003","url":null,"abstract":"Abstract An improved version of the Anshel–Anshel–Goldfeld (AAG) algebraic cryptographic key-exchange scheme, that is in particular resistant against the Tsaban linear span cryptanalysis, is established. Unlike the original version, that is based on the intractability of the simultaneous conjugacy search problem for the platform group, the proposed version is based on harder simultaneous membership-conjugacy search problems, and the membership problem needs to be solved for a subset of the platform group that can be easily and efficiently built to be very complicated and without any good structure. A number of other hard problems need to be solved first before start solving the simultaneous membership-conjugacy search problem to obtain the exchanged key.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"116 1","pages":"35 - 41"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76185655","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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