Abstract A recent collection of papers in the last years have given a renovated interest to the notion of orbit decidability. This is a new quite general algorithmic notion, connecting with several classical results, and closely related to the study of the conjugacy problem for extensions of groups. In the present survey we explain several of the classical results closely related to this concept, and we explain the main ideas behind the recent connection with the conjugacy problem made by Bogopolski–Martino–Ventura in [Trans. Amer. Math. Soc. 362 (2010), 2003–2036]. All the consequences up to date, published in several other papers by other authors, are also commented and reviewed.
{"title":"Group-theoretic orbit decidability","authors":"E. Ventura","doi":"10.1515/gcc-2014-0012","DOIUrl":"https://doi.org/10.1515/gcc-2014-0012","url":null,"abstract":"Abstract A recent collection of papers in the last years have given a renovated interest to the notion of orbit decidability. This is a new quite general algorithmic notion, connecting with several classical results, and closely related to the study of the conjugacy problem for extensions of groups. In the present survey we explain several of the classical results closely related to this concept, and we explain the main ideas behind the recent connection with the conjugacy problem made by Bogopolski–Martino–Ventura in [Trans. Amer. Math. Soc. 362 (2010), 2003–2036]. All the consequences up to date, published in several other papers by other authors, are also commented and reviewed.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"62 1","pages":"133 - 148"},"PeriodicalIF":0.0,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86011882","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 survey and reflect upon several aspects of the theory of infinite finitely generated and finitely presented groups that were originally motivated by work of Gilbert Baumslag. All but the last of the topics we have chosen are all related in one way or another to the theory of limit groups and the solution of the Tarski problems. These include the residually free and fully residually free properties and the big powers condition; Baumslag doubles and extensions of centralizers; residually-𝒳 groups and extensions of results of Benjamin Baumslag and finally the relationship between CT and CSA groups.
{"title":"Reflections on some aspects of infinite groups","authors":"B. Fine, A. Gaglione, G. Rosenberger, D. Spellman","doi":"10.1515/gcc-2014-0008","DOIUrl":"https://doi.org/10.1515/gcc-2014-0008","url":null,"abstract":"Abstract In this paper we survey and reflect upon several aspects of the theory of infinite finitely generated and finitely presented groups that were originally motivated by work of Gilbert Baumslag. All but the last of the topics we have chosen are all related in one way or another to the theory of limit groups and the solution of the Tarski problems. These include the residually free and fully residually free properties and the big powers condition; Baumslag doubles and extensions of centralizers; residually-𝒳 groups and extensions of results of Benjamin Baumslag and finally the relationship between CT and CSA groups.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":"81 - 91"},"PeriodicalIF":0.0,"publicationDate":"2014-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84133204","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 set of proper subgroups is a cover for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by σ ( G ) ${sigma(G)}$ . Determining σ ( G ) ${sigma(G)}$ is an open problem for many nonsolvable groups. For symmetric groups S n ${S_{n}}$ , Maróti determined σ ( S n ) ${sigma(S_{n})}$ for odd n with the exception of n = 9 ${n=9}$ and gave estimates for n even. In this paper we determine σ ( S n ) ${sigma(S_{n})}$ for n = 8 , 9 , 10 , 12 ${n=8,9,10,12}$ . In addition we find the covering number for the Mathieu group M 12 ${M_{12}}$ and improve an estimate given by Holmes for the Janko group J 1 ${J_{1}}$ .
{"title":"On the covering number of small symmetric groups and some sporadic simple groups","authors":"L. Kappe, Daniela Nikolova-Popova, Eric Swartz","doi":"10.1515/gcc-2016-0010","DOIUrl":"https://doi.org/10.1515/gcc-2016-0010","url":null,"abstract":"Abstract A set of proper subgroups is a cover for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by σ ( G ) ${sigma(G)}$ . Determining σ ( G ) ${sigma(G)}$ is an open problem for many nonsolvable groups. For symmetric groups S n ${S_{n}}$ , Maróti determined σ ( S n ) ${sigma(S_{n})}$ for odd n with the exception of n = 9 ${n=9}$ and gave estimates for n even. In this paper we determine σ ( S n ) ${sigma(S_{n})}$ for n = 8 , 9 , 10 , 12 ${n=8,9,10,12}$ . In addition we find the covering number for the Mathieu group M 12 ${M_{12}}$ and improve an estimate given by Holmes for the Janko group J 1 ${J_{1}}$ .","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"287 1","pages":"135 - 154"},"PeriodicalIF":0.0,"publicationDate":"2014-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73571181","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 propose a polynomial time quantum algorithm for solving the discrete logarithm problem (DLP) in matrices over finite group rings. The hardness of this problem was recently employed in the design of a key-exchange protocol proposed by D. Kahrobaei, C. Koupparis and V. Shpilrain [Groups Complex. Cryptol. 5 (2013), 97–115]. Our result implies that the Kahrobaei–Koupparis–Shpilrain protocol does not belong to the realm of post-quantum cryptography.
摘要提出了一种求解有限群环上矩阵离散对数问题的多项式时间量子算法。这个问题的难度最近被用于D. Kahrobaei, C. Koupparis和V. Shpilrain [Groups Complex]提出的密钥交换协议的设计中。密码学，5(2013)，97-115。我们的结果表明Kahrobaei-Koupparis-Shpilrain协议不属于后量子密码学领域。
{"title":"Quantum algorithm for discrete logarithm problem for matrices over finite group rings","authors":"A. Myasnikov, A. Ushakov","doi":"10.1515/gcc-2014-0003","DOIUrl":"https://doi.org/10.1515/gcc-2014-0003","url":null,"abstract":"Abstract. We propose a polynomial time quantum algorithm for solving the discrete logarithm problem (DLP) in matrices over finite group rings. The hardness of this problem was recently employed in the design of a key-exchange protocol proposed by D. Kahrobaei, C. Koupparis and V. Shpilrain [Groups Complex. Cryptol. 5 (2013), 97–115]. Our result implies that the Kahrobaei–Koupparis–Shpilrain protocol does not belong to the realm of post-quantum cryptography.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":"31 - 36"},"PeriodicalIF":0.0,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83755146","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 Given a group G and a G-module A, we show how to determine up to isomorphism the extensions E of A by G so that A embeds as smallest non-trivial term of the derived series or of the lower central series into E.
{"title":"Group extensions with special properties","authors":"A. Distler, B. Eick","doi":"10.1515/gcc-2015-0005","DOIUrl":"https://doi.org/10.1515/gcc-2015-0005","url":null,"abstract":"Abstract Given a group G and a G-module A, we show how to determine up to isomorphism the extensions E of A by G so that A embeds as smallest non-trivial term of the derived series or of the lower central series into E.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"61 1","pages":"1 - 10"},"PeriodicalIF":0.0,"publicationDate":"2014-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80629267","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 By a result of Gersten and Short finite presentations satisfying the usual non-metric small cancellation conditions present biautomatic groups. We show that in the case in which all pieces have length 1, a generalization of the C(3)-T(6) condition yields a larger collection of biautomatic groups.
{"title":"Generalized small cancellation presentations for automatic groups","authors":"R. Gilman","doi":"10.1515/gcc-2014-0007","DOIUrl":"https://doi.org/10.1515/gcc-2014-0007","url":null,"abstract":"Abstract By a result of Gersten and Short finite presentations satisfying the usual non-metric small cancellation conditions present biautomatic groups. We show that in the case in which all pieces have length 1, a generalization of the C(3)-T(6) condition yields a larger collection of biautomatic groups.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1978 1","pages":"101 - 93"},"PeriodicalIF":0.0,"publicationDate":"2014-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90267321","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}
Algorithms, constructions and examples are of central interest in combinatorial and geometric group theory. Teaching experience and, more recently, preparing a historical essay have led me to think the familiar group BS(1, 2) is an example of fundamental importance. The purpose of this note is to make a case for this point of view.We recall several interesting constructions and important examples of groups related to BS(1, 2), and indicate why certain of these groups played a key role in showing the word problem for nitely presented groups is unsolvable.
{"title":"Friends and relatives of BS(1,2)","authors":"C. F. Miller III","doi":"10.1515/gcc-2014-0006","DOIUrl":"https://doi.org/10.1515/gcc-2014-0006","url":null,"abstract":"Algorithms, constructions and examples are of central interest in combinatorial and geometric group theory. Teaching experience and, more recently, preparing a historical essay have led me to think the familiar group BS(1, 2) is an example of fundamental importance. The purpose of this note is to make a case for this point of view.We recall several interesting constructions and important examples of groups related to BS(1, 2), and indicate why certain of these groups played a key role in showing the word problem for nitely presented groups is unsolvable.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"32 1","pages":"73 - 80"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77527001","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 so-called Diophantine cryptology, a collection of cryptographic schemes where the computational security assumptions are based on hardness of solving some Diophantine equations, and some general ideas and techniques that occur in this area. In particular, we study an interesting variation of the endomorphism problem in groups, termed the double endomorphism problem. We prove that this problem is undecidable in free metabelian groups of sufficiently large rank. We relate this result to computational security assumptions of some group-based cryptosystems. In particular, we show how to improve the Grigoriev–Shpilrain's protocol to get a new computational security assumption based on the double endomorphism problem, providing a better theoretical foundation to security.
{"title":"Diophantine cryptography in free metabelian groups: Theoretical base","authors":"A. Myasnikov, V. Roman’kov","doi":"10.1515/gcc-2014-0011","DOIUrl":"https://doi.org/10.1515/gcc-2014-0011","url":null,"abstract":"Abstract In this paper we study so-called Diophantine cryptology, a collection of cryptographic schemes where the computational security assumptions are based on hardness of solving some Diophantine equations, and some general ideas and techniques that occur in this area. In particular, we study an interesting variation of the endomorphism problem in groups, termed the double endomorphism problem. We prove that this problem is undecidable in free metabelian groups of sufficiently large rank. We relate this result to computational security assumptions of some group-based cryptosystems. In particular, we show how to improve the Grigoriev–Shpilrain's protocol to get a new computational security assumption based on the double endomorphism problem, providing a better theoretical foundation to security.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"60 1","pages":"103 - 120"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84771729","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 survey, we discuss an emerging concept of decoy-based information security, or security without computational assumptions. In particular, we show how this concept can be implemented to provide security against (passive) computationally unbounded adversary in some public-key encryption protocols. In the world of symmetric cryptography, decoy-based security finds a wide range of applications, notably to secure delegation of computation to another party. We single out the scenario where a computationally limited party wants to send an encrypted message to a computationally superior party using the RSA protocol, thereby providing another kind of application of decoy ideas in a public-key setting. With typical RSA parameters, decoy-based method of delegation of computation improves the efficiency for the sender by several orders of magnitude.
{"title":"Decoy-based information security","authors":"V. Shpilrain","doi":"10.1515/gcc-2014-0010","DOIUrl":"https://doi.org/10.1515/gcc-2014-0010","url":null,"abstract":"Abstract In this survey, we discuss an emerging concept of decoy-based information security, or security without computational assumptions. In particular, we show how this concept can be implemented to provide security against (passive) computationally unbounded adversary in some public-key encryption protocols. In the world of symmetric cryptography, decoy-based security finds a wide range of applications, notably to secure delegation of computation to another party. We single out the scenario where a computationally limited party wants to send an encrypted message to a computationally superior party using the RSA protocol, thereby providing another kind of application of decoy ideas in a public-key setting. With typical RSA parameters, decoy-based method of delegation of computation improves the efficiency for the sender by several orders of magnitude.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"34 1","pages":"149 - 155"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75637955","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 group has finite palindromic width if there exists n such that every element can be expressed as a product of n or fewer palindromic words. We show that if G has finite palindromic width with respect to some generating set, then so does G≀ℤ r $G wr mathbb {Z}^{r}$ . We also give a new, self-contained proof that finitely generated metabelian groups have finite palindromic width. Finally, we show that solvable groups satisfying the maximal condition on normal subgroups (max-n) have finite palindromic width.
如果有n个元素可以表示为n个或更少的回文词的乘积，则群具有有限回文宽度。我们证明了如果G对于某个发电集具有有限的回文宽度，那么G献祭0 r $G wr mathbb {Z}^{r}$也是如此。我们还给出了有限生成的亚元群具有有限回文宽度的一个新的自包含证明。最后，我们证明了在正规子群(max-n)上满足极大条件的可解群具有有限的回文宽度。
{"title":"Palindromic width of wreath products, metabelian groups, and max-n solvable groups","authors":"T. Riley, Andrew W. Sale","doi":"10.1515/gcc-2014-0009","DOIUrl":"https://doi.org/10.1515/gcc-2014-0009","url":null,"abstract":"Abstract A group has finite palindromic width if there exists n such that every element can be expressed as a product of n or fewer palindromic words. We show that if G has finite palindromic width with respect to some generating set, then so does G≀ℤ r $G wr mathbb {Z}^{r}$ . We also give a new, self-contained proof that finitely generated metabelian groups have finite palindromic width. Finally, we show that solvable groups satisfying the maximal condition on normal subgroups (max-n) have finite palindromic width.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"124 1","pages":"121 - 132"},"PeriodicalIF":0.0,"publicationDate":"2013-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88004798","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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