L. Babinkostova, K. Bombardier, M. C. Cole, Thomas A. Morrell, Cory B. Scott
Abstract. We provide conditions under which the set of Rijndael-like functions considered as permutations of the state space and based on operations of the finite field GF (p k )${mathrm {GF}(p^k)}$ ( p≥2${pge 2}$ ) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the Advanced Encryption Standard (AES), a Rijndael cipher with specific block sizes. In [Discrete Appl. Math. 156 (2008), 3139–3149], R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF (2 k )${mathrm {GF}(2^k)}$ is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF (p k )${mathrm {GF}(p^k)}$ ( p≥2${pge 2}$ ) is equal to the symmetric group or the alternating group on the state space.
{"title":"Algebraic properties of generalized Rijndael-like ciphers","authors":"L. Babinkostova, K. Bombardier, M. C. Cole, Thomas A. Morrell, Cory B. Scott","doi":"10.1515/gcc-2014-0004","DOIUrl":"https://doi.org/10.1515/gcc-2014-0004","url":null,"abstract":"Abstract. We provide conditions under which the set of Rijndael-like functions considered as permutations of the state space and based on operations of the finite field GF (p k )${mathrm {GF}(p^k)}$ ( p≥2${pge 2}$ ) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the Advanced Encryption Standard (AES), a Rijndael cipher with specific block sizes. In [Discrete Appl. Math. 156 (2008), 3139–3149], R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF (2 k )${mathrm {GF}(2^k)}$ is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF (p k )${mathrm {GF}(p^k)}$ ( p≥2${pge 2}$ ) is equal to the symmetric group or the alternating group on the state space.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"24 8 1","pages":"37 - 54"},"PeriodicalIF":0.0,"publicationDate":"2012-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91059250","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 describe an algorithm that, on input of a recursive presentation P of a group, outputs a recursive presentation of a torsion-free quotient of P, isomorphic to P whenever P is itself torsion-free. Using this, we show the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed (first proved by Belegradek). We apply our techniques to show that recognising embeddability of finitely presented groups is Π 2 0 $Pi ^{0}_{2}$ -hard, Σ 2 0 $Sigma ^{0}_{2}$ -hard, and lies in Σ 3 0 $Sigma ^{0}_{3}$ . We also show that the sets of orders of torsion elements of finitely presented groups are precisely the Σ 2 0 $Sigma ^{0}_{2}$ sets which are closed under taking factors.
{"title":"On torsion in finitely presented groups","authors":"Maurice Chiodo","doi":"10.1515/gcc-2014-0001","DOIUrl":"https://doi.org/10.1515/gcc-2014-0001","url":null,"abstract":"Abstract. We describe an algorithm that, on input of a recursive presentation P of a group, outputs a recursive presentation of a torsion-free quotient of P, isomorphic to P whenever P is itself torsion-free. Using this, we show the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed (first proved by Belegradek). We apply our techniques to show that recognising embeddability of finitely presented groups is Π 2 0 $Pi ^{0}_{2}$ -hard, Σ 2 0 $Sigma ^{0}_{2}$ -hard, and lies in Σ 3 0 $Sigma ^{0}_{3}$ . We also show that the sets of orders of torsion elements of finitely presented groups are precisely the Σ 2 0 $Sigma ^{0}_{2}$ sets which are closed under taking factors.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"24 1","pages":"1 - 8"},"PeriodicalIF":0.0,"publicationDate":"2011-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80177211","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. This is the second paper in a series of four, where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Here, for an arbitrary group G of infinite words over an ordered abelian group Λ we construct a Λ-tree Γ G $Gamma _G$ equipped with a free action of G. Moreover, we show that Γ G $Gamma _G$ is a universal tree for G in the sense that it isometrically and equivariantly embeds into every Λ-tree equipped with a free G-action compatible with the original length function on G. Also, for a group G acting freely on a Λ-tree Γ we show how one can easily obtain an embedding of G into the set of reduced infinite words R(Λ,X)$R(Lambda , X)$ , where the alphabet X is obtained from the action G on Γ.
摘要这是四篇系列论文中的第二篇，我们将讨论非阿基米德群作用，长度函数和无限词的统一理论。在这里，对于有序阿贝尔群Λ上的任意无限词群G，我们构造了一个具有自由G作用的Λ-tree Γ G $Gamma _G$，并且我们证明了Γ G $Gamma _G$是G的一棵泛树，因为它等距地、等距地嵌入到每一个具有与G上的原始长度函数兼容的自由G作用的Λ-tree中。对于自由作用于Λ-tree Γ上的群G，我们展示了如何容易地将G嵌入到约简无限词集R(Λ，X) $R(Lambda , X)$中，其中字母X是由作用于Γ上的G获得的。
{"title":"Infinite words and universal free actions","authors":"O. Kharlampovich, A. Myasnikov, Denis Serbin","doi":"10.1515/gcc-2014-0005","DOIUrl":"https://doi.org/10.1515/gcc-2014-0005","url":null,"abstract":"Abstract. This is the second paper in a series of four, where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Here, for an arbitrary group G of infinite words over an ordered abelian group Λ we construct a Λ-tree Γ G $Gamma _G$ equipped with a free action of G. Moreover, we show that Γ G $Gamma _G$ is a universal tree for G in the sense that it isometrically and equivariantly embeds into every Λ-tree equipped with a free G-action compatible with the original length function on G. Also, for a group G acting freely on a Λ-tree Γ we show how one can easily obtain an embedding of G into the set of reduced infinite words R(Λ,X)$R(Lambda , X)$ , where the alphabet X is obtained from the action G on Γ.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"170 1","pages":"55 - 69"},"PeriodicalIF":0.0,"publicationDate":"2011-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83367132","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 class of groups whose word problem is poly-context-free; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free groups has poly-context-free word problem, and conjecture that the converse also holds. We prove our conjecture for several classes of soluble groups, including metabelian groups and torsion-free soluble groups, and present progress towards resolving the conjecture for soluble groups in general. Some of the techniques introduced for proving languages not to be poly-context-free may be of independent interest.
{"title":"Groups with poly-context-free word problem","authors":"Tara Brough","doi":"10.1515/gcc-2014-0002","DOIUrl":"https://doi.org/10.1515/gcc-2014-0002","url":null,"abstract":"Abstract. We consider the class of groups whose word problem is poly-context-free; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free groups has poly-context-free word problem, and conjecture that the converse also holds. We prove our conjecture for several classes of soluble groups, including metabelian groups and torsion-free soluble groups, and present progress towards resolving the conjecture for soluble groups in general. Some of the techniques introduced for proving languages not to be poly-context-free may be of independent interest.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"36 1","pages":"29 - 9"},"PeriodicalIF":0.0,"publicationDate":"2011-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84713154","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 algorithm which computes a presentation for a subgroup, denoted 𝒜ℳ g,p,1 ${mathcal {AM}_{g,p,1}}$ , of the automorphism group of a free group. It is known that 𝒜ℳ g,p,1 ${mathcal {AM}_{g,p,1}}$ is isomorphic to the mapping class group of an orientable genus-g surface with p punctures and one boundary component. We define a variation of the Auter space.
{"title":"A combinatorial algorithm to compute presentations of mapping class groups of orientable surfaces with one boundary component","authors":"Lluís Bacardit","doi":"10.1515/gcc-2015-0011","DOIUrl":"https://doi.org/10.1515/gcc-2015-0011","url":null,"abstract":"Abstract We give an algorithm which computes a presentation for a subgroup, denoted 𝒜ℳ g,p,1 ${mathcal {AM}_{g,p,1}}$ , of the automorphism group of a free group. It is known that 𝒜ℳ g,p,1 ${mathcal {AM}_{g,p,1}}$ is isomorphic to the mapping class group of an orientable genus-g surface with p punctures and one boundary component. We define a variation of the Auter space.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"74 1","pages":"115 - 95"},"PeriodicalIF":0.0,"publicationDate":"2010-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76554804","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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