Iris Anshel, Derek Atkins, D. Goldfeld, P. Gunnells
Abstract This paper introduces a novel braid based cryptographic hash function candidate which is suitable for use in low resource environments. It is shown that the new hash function performed extremely well on a range of cryptographic test suites.
{"title":"A class of hash functions based on the algebraic eraser™","authors":"Iris Anshel, Derek Atkins, D. Goldfeld, P. Gunnells","doi":"10.1515/gcc-2016-0004","DOIUrl":"https://doi.org/10.1515/gcc-2016-0004","url":null,"abstract":"Abstract This paper introduces a novel braid based cryptographic hash function candidate which is suitable for use in low resource environments. It is shown that the new hash function performed extremely well on a range of cryptographic test suites.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"33 1","pages":"1 - 7"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83643975","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 paper introduces a new type of attack, termed a nonlinear decomposition attack, against two known group-based key agreement protocols, namely, protocol based on extensions of (semi)groups by endomorphisms introduced by Kahrobaei, Shpilrain et al., and the noncommutative Diffie–Hellman protocol introduced by Ko, Lee et al. This attack works efficiently in the case when finitely generated nilpotent (more generally, polycyclic) groups are used as platforms. This attack is based on a deterministic algorithm that finds the secret shared key from the public data in both the protocols under consideration. Furthermore, we show that in this case one can break the schemes without solving the algorithmic problems on which the assumptions are based. The efficacy of the attack depends on the platform group, so it requires a more thorough analysis in each particular case.
{"title":"A nonlinear decomposition attack","authors":"V. Roman’kov","doi":"10.1515/gcc-2016-0017","DOIUrl":"https://doi.org/10.1515/gcc-2016-0017","url":null,"abstract":"Abstract This paper introduces a new type of attack, termed a nonlinear decomposition attack, against two known group-based key agreement protocols, namely, protocol based on extensions of (semi)groups by endomorphisms introduced by Kahrobaei, Shpilrain et al., and the noncommutative Diffie–Hellman protocol introduced by Ko, Lee et al. This attack works efficiently in the case when finitely generated nilpotent (more generally, polycyclic) groups are used as platforms. This attack is based on a deterministic algorithm that finds the secret shared key from the public data in both the protocols under consideration. Furthermore, we show that in this case one can break the schemes without solving the algorithmic problems on which the assumptions are based. The efficacy of the attack depends on the platform group, so it requires a more thorough analysis in each particular case.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"34 1","pages":"197 - 207"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79876598","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 A semiring is an algebraic structure satisfying the usual axioms for a not necessarily commutative ring, but without the requirement that addition be invertible. Aside from rings, well-studied instances in cryptographic applications include the Boolean semiring and the tropical semiring. The latter, in particular, behaves to a large extent like a field and exhibits interesting properties in the cryptographic context. This short note explores a GPU-based highly parallel implementation of a protocol recently proposed by Grigoriev and Shpilrain [Comm. Algebra 42 (2014), 2624–2632], in the context of Diffie–Hellman key agreements.
{"title":"Key agreement under tropical parallels","authors":"J. Chauvet, É. Mahé","doi":"10.1515/gcc-2015-0013","DOIUrl":"https://doi.org/10.1515/gcc-2015-0013","url":null,"abstract":"Abstract A semiring is an algebraic structure satisfying the usual axioms for a not necessarily commutative ring, but without the requirement that addition be invertible. Aside from rings, well-studied instances in cryptographic applications include the Boolean semiring and the tropical semiring. The latter, in particular, behaves to a large extent like a field and exhibits interesting properties in the cryptographic context. This short note explores a GPU-based highly parallel implementation of a protocol recently proposed by Grigoriev and Shpilrain [Comm. Algebra 42 (2014), 2624–2632], in the context of Diffie–Hellman key agreements.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"20 1","pages":"195 - 198"},"PeriodicalIF":0.0,"publicationDate":"2015-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87915976","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 a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).
{"title":"Public-key cryptosystem based on invariants of diagonalizable groups","authors":"M. Jurás, F. Marko, A. Zubkov","doi":"10.1515/gcc-2017-0003","DOIUrl":"https://doi.org/10.1515/gcc-2017-0003","url":null,"abstract":"Abstract We develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"151 1","pages":"31 - 54"},"PeriodicalIF":0.0,"publicationDate":"2015-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88872899","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 2013, Kharlampovich, Myasnikov, and Sapir constructed the first examples of finitely presented residually finite groups with large Dehn functions. Given any recursive function f, they produce a finitely presented residually finite group with Dehn function dominating f. There are no known elementary examples of finitely presented residually finite groups with super-exponential Dehn function. Dison and Riley’s hydra groups can be used to construct a sequence of groups for which the Dehn function of the kth group is equivalent to the kth Ackermann function. Kharlampovich, Myasnikov, and Sapir asked whether or not these groups are residually finite. We show that these constructions do not produce residually finite groups.
{"title":"Hydra group doubles are not residually finite","authors":"K. Pueschel","doi":"10.1515/gcc-2016-0015","DOIUrl":"https://doi.org/10.1515/gcc-2016-0015","url":null,"abstract":"Abstract In 2013, Kharlampovich, Myasnikov, and Sapir constructed the first examples of finitely presented residually finite groups with large Dehn functions. Given any recursive function f, they produce a finitely presented residually finite group with Dehn function dominating f. There are no known elementary examples of finitely presented residually finite groups with super-exponential Dehn function. Dison and Riley’s hydra groups can be used to construct a sequence of groups for which the Dehn function of the kth group is equivalent to the kth Ackermann function. Kharlampovich, Myasnikov, and Sapir asked whether or not these groups are residually finite. We show that these constructions do not produce residually finite groups.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"87 1","pages":"163 - 170"},"PeriodicalIF":0.0,"publicationDate":"2015-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86728200","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 Let G be a free metabelian group of rank r = 2. We introduce a faithful 2×2 real matrix representation of G and extend this to a group G ℤ[θ] $G^{mathbb {Z}[theta ]}$ of 2×2 matrices admitting exponents from the integral polynomial ring ℤ[θ]$mathbb {Z}[theta ]$ . Identifying G with its matrix representation, we show that given γ(θ)∈G ℤ[θ] $gamma (theta )in G^{mathbb {Z}[theta ]}$ and n∈ℤ$nin mathbb {Z}$ , one has that lim θ→n γ(θ)$lim _{theta rightarrow n}gamma (theta )$ exists and lies in G. Furthermore, the maps γ(θ)↦lim θ→n γ(θ)$gamma (theta )mapsto lim _{theta rightarrow n}gamma (theta )$ form a discriminating family of group retractions G ℤ[θ] →G$G^{mathbb {Z}[theta ]}rightarrow G$ as n varies over ℤ. Although not explicitly carried out in this manuscript, it is clear that similar results hold for any countable rank r.
{"title":"An application of elementary real analysis to a metabelian group admitting integral polynomial exponents","authors":"A. Gaglione, S. Lipschutz, D. Spellman","doi":"10.1515/gcc-2015-0004","DOIUrl":"https://doi.org/10.1515/gcc-2015-0004","url":null,"abstract":"Abstract Let G be a free metabelian group of rank r = 2. We introduce a faithful 2×2 real matrix representation of G and extend this to a group G ℤ[θ] $G^{mathbb {Z}[theta ]}$ of 2×2 matrices admitting exponents from the integral polynomial ring ℤ[θ]$mathbb {Z}[theta ]$ . Identifying G with its matrix representation, we show that given γ(θ)∈G ℤ[θ] $gamma (theta )in G^{mathbb {Z}[theta ]}$ and n∈ℤ$nin mathbb {Z}$ , one has that lim θ→n γ(θ)$lim _{theta rightarrow n}gamma (theta )$ exists and lies in G. Furthermore, the maps γ(θ)↦lim θ→n γ(θ)$gamma (theta )mapsto lim _{theta rightarrow n}gamma (theta )$ form a discriminating family of group retractions G ℤ[θ] →G$G^{mathbb {Z}[theta ]}rightarrow G$ as n varies over ℤ. Although not explicitly carried out in this manuscript, it is clear that similar results hold for any countable rank r.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"13 1","pages":"59 - 68"},"PeriodicalIF":0.0,"publicationDate":"2015-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84531997","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 explore a method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn and use it to show that with respect to the usual action of Fm×ℤn on Tree ×ℝ n ${mathrm {Tree}times mathbb {R}^n}$ , every quasiconvex subgroup of Fm×ℤn is convex. Further, we show that the Cartan–Hadamard theorem can be used to show that locally convex subsets of complete and connected CAT(0) spaces are convex. Finally, we show that the quasiconvex subgroups of Fm×ℤn are precisely those of the form A×B, where A≤F m ${Ale F_m}$ is finitely generated, and B≤ℤ n ${Ble mathbb {Z}^n}$ .
{"title":"On convex hulls and the quasiconvex subgroups of Fm×ℤn","authors":"Jordan Sahattchieve","doi":"10.1515/gcc-2015-0006","DOIUrl":"https://doi.org/10.1515/gcc-2015-0006","url":null,"abstract":"Abstract In this paper, we explore a method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn and use it to show that with respect to the usual action of Fm×ℤn on Tree ×ℝ n ${mathrm {Tree}times mathbb {R}^n}$ , every quasiconvex subgroup of Fm×ℤn is convex. Further, we show that the Cartan–Hadamard theorem can be used to show that locally convex subsets of complete and connected CAT(0) spaces are convex. Finally, we show that the quasiconvex subgroups of Fm×ℤn are precisely those of the form A×B, where A≤F m ${Ale F_m}$ is finitely generated, and B≤ℤ n ${Ble mathbb {Z}^n}$ .","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"17 1","pages":"69 - 80"},"PeriodicalIF":0.0,"publicationDate":"2015-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82313848","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 A finite symmetric graph Γ is a pair (Γ,f)$(Gamma ,f)$ , where Γ is a finite graph and f:Γ→Γ$f:Gamma rightarrow Gamma $ is a graph self equivalence or automorphism. We develop several tools for studying such symmetries. In particular, we describe in detail all symmetries with a single edge orbit, we prove that each symmetric graph has a maximal forest that meets each edge orbit in a sequential set of edges – a sequential maximal forest – and we calculate the characteristic polynomial χ f (t)$chi _f(t)$ and the minimal polynomial μ f (t)$mu _f(t)$ of the linear map H 1 (f):H 1 (Γ,ℤ)→H 1 (Γ,ℤ)$H_1(f):H_1(Gamma ,mathbb {Z})rightarrow H_1(Gamma ,mathbb {Z})$ . The calculation is in terms of the quotient graph Γ ¯$overline{Gamma }$ .
{"title":"Symmetries of finite graphs and homology","authors":"Benjamin Atchison, E. Turner","doi":"10.1515/gcc-2015-0003","DOIUrl":"https://doi.org/10.1515/gcc-2015-0003","url":null,"abstract":"Abstract A finite symmetric graph Γ is a pair (Γ,f)$(Gamma ,f)$ , where Γ is a finite graph and f:Γ→Γ$f:Gamma rightarrow Gamma $ is a graph self equivalence or automorphism. We develop several tools for studying such symmetries. In particular, we describe in detail all symmetries with a single edge orbit, we prove that each symmetric graph has a maximal forest that meets each edge orbit in a sequential set of edges – a sequential maximal forest – and we calculate the characteristic polynomial χ f (t)$chi _f(t)$ and the minimal polynomial μ f (t)$mu _f(t)$ of the linear map H 1 (f):H 1 (Γ,ℤ)→H 1 (Γ,ℤ)$H_1(f):H_1(Gamma ,mathbb {Z})rightarrow H_1(Gamma ,mathbb {Z})$ . The calculation is in terms of the quotient graph Γ ¯$overline{Gamma }$ .","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"102 1","pages":"11 - 30"},"PeriodicalIF":0.0,"publicationDate":"2015-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78174503","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 to decide whether a given braid with four strings is a product of three factors which are conjugates of standard generators of the braid group. The algorithm is of polynomial time. It is based on the Garside theory. We give also a polynomial algorithm to decide if a given braid with any number of strings is a product of two factors which are conjugates of given powers of the standard generators (in my previous paper this problem was solved without polynomial estimates).
{"title":"Algorithmic recognition of quasipositive 4-braids of algebraic length three","authors":"S. Orevkov","doi":"10.1515/gcc-2015-0012","DOIUrl":"https://doi.org/10.1515/gcc-2015-0012","url":null,"abstract":"Abstract We give an algorithm to decide whether a given braid with four strings is a product of three factors which are conjugates of standard generators of the braid group. The algorithm is of polynomial time. It is based on the Garside theory. We give also a polynomial algorithm to decide if a given braid with any number of strings is a product of two factors which are conjugates of given powers of the standard generators (in my previous paper this problem was solved without polynomial estimates).","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"26 1","pages":"157 - 173"},"PeriodicalIF":0.0,"publicationDate":"2015-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89752754","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 is not known whether Thompson's group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to 𝒞-graph automatic by the authors, a compelling question is whether F is graph automatic or 𝒞-graph automatic for an appropriate language class 𝒞. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a 1-counter graph automatic structure for F based on the standard infinite normal form for group elements.
{"title":"Thompson's group F is 1-counter graph automatic","authors":"M. Elder, J. Taback","doi":"10.1515/gcc-2016-0001","DOIUrl":"https://doi.org/10.1515/gcc-2016-0001","url":null,"abstract":"Abstract It is not known whether Thompson's group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to 𝒞-graph automatic by the authors, a compelling question is whether F is graph automatic or 𝒞-graph automatic for an appropriate language class 𝒞. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a 1-counter graph automatic structure for F based on the standard infinite normal form for group elements.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"113 1","pages":"21 - 33"},"PeriodicalIF":0.0,"publicationDate":"2015-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73940623","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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