Abstract Recently, Shpilrain and Sosnovski proposed a hash function based on composition of affine maps. In this paper, we show that this hash function with its proposed parameters is not weak collision resistant, for plaintexts of size at least 1.9MB (about 2 24 {2^{24}} bits). Our approach is to reduce the preimage problem to a (very) high density instance of the Random Modular Subset Sum Problem, for which we give an algorithm capable of solving instances of the resulting size. Specifically, given plaintexts of about 1.9MB, we were able to produce other plaintexts of the same size with the same hash value in about 13 hours each, on average.
{"title":"Cryptanalysis of a hash function, and the modular subset sum problem","authors":"C. Monico","doi":"10.1515/gcc-2019-2001","DOIUrl":"https://doi.org/10.1515/gcc-2019-2001","url":null,"abstract":"Abstract Recently, Shpilrain and Sosnovski proposed a hash function based on composition of affine maps. In this paper, we show that this hash function with its proposed parameters is not weak collision resistant, for plaintexts of size at least 1.9MB (about 2 24 {2^{24}} bits). Our approach is to reduce the preimage problem to a (very) high density instance of the Random Modular Subset Sum Problem, for which we give an algorithm capable of solving instances of the resulting size. Specifically, given plaintexts of about 1.9MB, we were able to produce other plaintexts of the same size with the same hash value in about 13 hours each, on average.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"17 1","pages":"17 - 23"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77009205","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 a finite group G, let π e ( G ) {pi_{e}(G)} be the set of orders of elements of G, let s k {s_{k}} denote the number of elements of order k in G, for each k ∈ π e ( G ) {kinpi_{e}(G)} , and then let nse ( G ) {operatorname{nse}(G)} be the unordered set { s k : k ∈ π e ( G ) } {{s_{k}:kinpi_{e}(G)}} . In this paper, it is shown that if | G | = | L 2 ( q ) | {lvert Grvert=lvert L_{2}(q)rvert} and nse ( G ) = nse ( L 2 ( q ) ) {operatorname{nse}(G)=operatorname{nse}(L_{2}(q))} for some prime-power q, then G is isomorphic to L 2 ( q ) {L_{2}(q)} .
{"title":"Recognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each order","authors":"Alireza Khalili Asboei","doi":"10.1515/gcc-2018-0011","DOIUrl":"https://doi.org/10.1515/gcc-2018-0011","url":null,"abstract":"Abstract In a finite group G, let π e ( G ) {pi_{e}(G)} be the set of orders of elements of G, let s k {s_{k}} denote the number of elements of order k in G, for each k ∈ π e ( G ) {kinpi_{e}(G)} , and then let nse ( G ) {operatorname{nse}(G)} be the unordered set { s k : k ∈ π e ( G ) } {{s_{k}:kinpi_{e}(G)}} . In this paper, it is shown that if | G | = | L 2 ( q ) | {lvert Grvert=lvert L_{2}(q)rvert} and nse ( G ) = nse ( L 2 ( q ) ) {operatorname{nse}(G)=operatorname{nse}(L_{2}(q))} for some prime-power q, then G is isomorphic to L 2 ( q ) {L_{2}(q)} .","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"12 1","pages":"111 - 118"},"PeriodicalIF":0.0,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79527366","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 introduce two general schemes of algebraic cryptography. We show that many of the systems and protocols considered in literature that use two-sided multiplications are specific cases of the first general scheme. In a similar way, we introduce the second general scheme that joins systems and protocols based on automorphisms or endomorphisms of algebraic systems. Also, we discuss possible applications of the membership search problem in algebraic cryptanalysis. We show how an efficient decidability of the underlined membership search problem for an algebraic system chosen as the platform can be applied to show a vulnerability of both schemes. Our attacks are based on the linear or on the nonlinear decomposition method, which complete each other. We give a couple of examples of systems and protocols known in the literature that use one of the two introduced schemes with their cryptanalysis. Mostly, these protocols simulate classical cryptographic schemes, such as Diffie–Hellman, Massey–Omura and ElGamal in algebraic setting. Furthermore, we show that, in many cases, one can break the schemes without solving the algorithmic problems on which the assumptions are based.
{"title":"Two general schemes of algebraic cryptography","authors":"V. Roman’kov","doi":"10.1515/gcc-2018-0009","DOIUrl":"https://doi.org/10.1515/gcc-2018-0009","url":null,"abstract":"Abstract In this paper, we introduce two general schemes of algebraic cryptography. We show that many of the systems and protocols considered in literature that use two-sided multiplications are specific cases of the first general scheme. In a similar way, we introduce the second general scheme that joins systems and protocols based on automorphisms or endomorphisms of algebraic systems. Also, we discuss possible applications of the membership search problem in algebraic cryptanalysis. We show how an efficient decidability of the underlined membership search problem for an algebraic system chosen as the platform can be applied to show a vulnerability of both schemes. Our attacks are based on the linear or on the nonlinear decomposition method, which complete each other. We give a couple of examples of systems and protocols known in the literature that use one of the two introduced schemes with their cryptanalysis. Mostly, these protocols simulate classical cryptographic schemes, such as Diffie–Hellman, Massey–Omura and ElGamal in algebraic setting. Furthermore, we show that, in many cases, one can break the schemes without solving the algorithmic problems on which the assumptions are based.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"20 1","pages":"83 - 98"},"PeriodicalIF":0.0,"publicationDate":"2018-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80824720","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 Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most C⋅LN-2{Ccdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance.
{"title":"Garside theory and subsurfaces: Some examples in braid groups","authors":"S. Schleimer, B. Wiest","doi":"10.1515/gcc-2019-2007","DOIUrl":"https://doi.org/10.1515/gcc-2019-2007","url":null,"abstract":"Abstract Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most C⋅LN-2{Ccdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"119 1","pages":"61 - 75"},"PeriodicalIF":0.0,"publicationDate":"2018-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79267070","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 prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.
{"title":"On finitely generated submonoids of virtually free groups","authors":"Pedro V. Silva, A. Zakharov","doi":"10.1515/gcc-2018-0008","DOIUrl":"https://doi.org/10.1515/gcc-2018-0008","url":null,"abstract":"Abstract We prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"43 1","pages":"63 - 82"},"PeriodicalIF":0.0,"publicationDate":"2018-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74186401","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 Suppose that G is a finitely generated group and WP ( G ) {operatorname{WP}(G)} is the formal language of words defining the identity in G. We prove that if G is a virtually nilpotent group that is not virtually abelian, the fundamental group of a finite volume hyperbolic three-manifold, or a right-angled Artin group whose graph lies in a certain infinite class, then WP ( G ) {operatorname{WP}(G)} is not a multiple context-free language.
{"title":"Groups whose word problems are not semilinear","authors":"R. Gilman, Robert P. Kropholler, S. Schleimer","doi":"10.1515/gcc-2018-0010","DOIUrl":"https://doi.org/10.1515/gcc-2018-0010","url":null,"abstract":"Abstract Suppose that G is a finitely generated group and WP ( G ) {operatorname{WP}(G)} is the formal language of words defining the identity in G. We prove that if G is a virtually nilpotent group that is not virtually abelian, the fundamental group of a finite volume hyperbolic three-manifold, or a right-angled Artin group whose graph lies in a certain infinite class, then WP ( G ) {operatorname{WP}(G)} is not a multiple context-free language.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"21 1","pages":"53 - 62"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77943612","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 each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that 𝒦 {{mathcal{K}}} , the class of groups which satisfy the conjecture, is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in 𝒦 {{mathcal{K}}} or more generally two torsion-free groups are universally equivalent.
{"title":"Orderable groups, elementary theory, and the Kaplansky conjecture","authors":"B. Fine, A. Gaglione, G. Rosenberger, D. Spellman","doi":"10.1515/gcc-2018-0005","DOIUrl":"https://doi.org/10.1515/gcc-2018-0005","url":null,"abstract":"Abstract We show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that 𝒦 {{mathcal{K}}} , the class of groups which satisfy the conjecture, is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in 𝒦 {{mathcal{K}}} or more generally two torsion-free groups are universally equivalent.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"31 1","pages":"43 - 52"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81694530","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 a practical fully homomorphic encryption (FHE) scheme based on homomorphisms between rings and show that it enables very efficient computation on encrypted data. Our encryption though is private-key; public information is only used to operate on encrypted data without decrypting it. Still, we show that our method allows for a third party search on encrypted data.
{"title":"Practical private-key fully homomorphic encryption in rings","authors":"A. Gribov, Delaram Kahrobaei, V. Shpilrain","doi":"10.1515/gcc-2018-0006","DOIUrl":"https://doi.org/10.1515/gcc-2018-0006","url":null,"abstract":"Abstract We describe a practical fully homomorphic encryption (FHE) scheme based on homomorphisms between rings and show that it enables very efficient computation on encrypted data. Our encryption though is private-key; public information is only used to operate on encrypted data without decrypting it. Still, we show that our method allows for a third party search on encrypted data.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"36 1","pages":"17 - 27"},"PeriodicalIF":0.0,"publicationDate":"2018-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74379043","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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