{"title":"Metabelian Product of a Free Nilpotent Group with a Free Abelian Group","authors":"M. H. Dean","doi":"10.1515/GCC.2009.169","DOIUrl":"https://doi.org/10.1515/GCC.2009.169","url":null,"abstract":"If two groups are residually-𝑃, their free product is not necessarily so; however, it is known that the free product of residually torsion-free nilpotent groups is again residually torsion-free nilpotent. In this paper it is shown that the free metabelian product of a free nilpotent group of class two with a free abelian group is residually torsion-free nilpotent.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"49 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125918994","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":"Space Complexity and Word Problems of Groups","authors":"S. R. Lakin, R. Thomas","doi":"10.1515/GCC.2009.261","DOIUrl":"https://doi.org/10.1515/GCC.2009.261","url":null,"abstract":"This paper is concerned with the question of determining which groups have their word problems lying in a given complexity class. Our main results give sufficient conditions for the existence of groups whose word problem is contained in some specified space complexity class but is not contained in some other given space complexity class.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"139 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122468398","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":"No-leak authentication by the Sherlock Holmes method","authors":"D. Grigoriev, V. Shpilrain","doi":"10.1515/gcc-2012-0009","DOIUrl":"https://doi.org/10.1515/gcc-2012-0009","url":null,"abstract":"Abstract. We propose a class of authentication schemes that are literally zero-knowledge, as compared to what is formally defined as “zero-knowledge” in cryptographic literature. We call this “no-leak” authentication to distinguish from an established “zero-knowledge” concept. The “no-leak” condition implies “zero-knowledge” (even “perfect zero-knowledge”), but it is actually stronger, as we illustrate by examples. The principal idea behind our schemes is: the verifier challenges the prover with questions that he (the verifier) already knows answers to; therefore, even a computationally unbounded verifier who follows the protocol cannot possibly learn anything new during any number of authentication sessions. This is therefore also true for a computationally unbounded passive adversary.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134171543","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":"On asymptotic densities and generic properties in finitely generated groups","authors":"C. Carstensen, B. Fine, G. Rosenberger","doi":"10.1515/gcc.2010.008","DOIUrl":"https://doi.org/10.1515/gcc.2010.008","url":null,"abstract":"Abstract Recall that asymptotic density is a method to compute densities and/or probabilities within infinite finitely generated groups. If is a group property, the asymptotic density determines the measure of the set of elements which satisfy . Is this asymptotic density equal to 1, we say that the property is generic in G. is called an asymptotic visible property, if the corresponding asymptotic density is strictly between 0 and 1. If the asymptotic density is 0, then is called negligible. We call a group property suitable if it is preserved under isomorphisms and its asymptotic density exists and is independent of finite generating systems. In this paper we prove that there is an interesting connection between the strong generic free group property of a group G and its subgroups of finite index.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124194787","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":"Subgroup conjugacy problem for Garside subgroups of Garside groups","authors":"A. Kalka, E. Liberman, M. Teicher","doi":"10.1515/gcc.2010.010","DOIUrl":"https://doi.org/10.1515/gcc.2010.010","url":null,"abstract":"Abstract We solve the subgroup conjugacy problem for parabolic subgroups and Garside subgroups of a Garside group, and we present deterministic algorithms. This solution may be improved by using minimal simple elements. For standard parabolic subgroups of Garside groups we provide effective algorithms for computing minimal simple elements.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131411791","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":"A new generic digital signature algorithm","authors":"J. Seberry, V. To, Dongvu Tonien","doi":"10.1515/gcc.2011.008","DOIUrl":"https://doi.org/10.1515/gcc.2011.008","url":null,"abstract":"Abstract In this paper, we study two digital signature algorithms, the DSA and ECDSA, which have become NIST standard and have been widely used in almost all commercial applications. We will show that the two algorithms are actually ‘the same’ algebraically and propose a generic algorithm such that both DSA and ECDSA are instances of it. By looking at this special angle through the generic algorithm, we gain a new insight into the two algorithms DSA and ECDSA. Our new proposed digital signature algorithm is described generically using a group G and a map to Number : G → ℤ. As an illustration, we choose G to be a group of non-singular circulant matrices over finite field and describe a totally new concrete digital signature algorithm.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130676993","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":"Random van Kampen diagrams and algorithmic problems in groups","authors":"A. Myasnikov, A. Ushakov","doi":"10.1515/gcc.2011.006","DOIUrl":"https://doi.org/10.1515/gcc.2011.006","url":null,"abstract":"Abstract In this paper we study the structure of random van Kampen diagrams over finitely presented groups. Such diagrams have many remarkable properties. In particular, we show that a random van Kampen diagram over a given group is hyperbolic, even though the group itself may not be hyperbolic. This allows one to design new fast algorithms for the Word Problem in groups. We introduce and study a new filling function, the depth of van Kampen diagrams, – a crucial algorithmic characteristic of null-homotopic words in the group.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121959409","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":"A note on the homology of hyperbolic groups","authors":"Matthias Neumann-Brosig, G. Rosenberger","doi":"10.1515/gcc.2010.012","DOIUrl":"https://doi.org/10.1515/gcc.2010.012","url":null,"abstract":"Abstract Hyperbolic groups have been studied in various fields in mathematics. They appear in contexts as diverse as geometric group theory, function theory (as Fuchsian groups) and algebraic topology (as fundamental groups of compact hyperbolic surfaces). Hyperbolic groups possess geometrical properties well suited for the study of homological finiteness conditions. In this paper we will prove some of these results via free resolutions obtained from the Rips-complex.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114949582","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":"Almost Locally Free Groups and a Theorem of Magnus: Some Questions","authors":"A. Gaglione, S. Lipschutz, D. Spellman","doi":"10.1515/GCC.2009.181","DOIUrl":"https://doi.org/10.1515/GCC.2009.181","url":null,"abstract":"Ben Fine observed that a theorem of Magnus on normal closures of elements in free groups is first order expressible and thus holds in every elementarily free group. This classical theorem, vintage 1931, asserts that if two elements in a free group have the same normal closure, then either they are conjugate or one is conjugate to the inverse of the other in the free group. An examination of a set of sentences capturing this theorem reveals that the sentences are universal-existential. Consequently the theorem holds in the almost locally free groups of Gaglione and Spellman. We give a sufficient condition for the theorem to hold in every fully residually free group as well as a sufficient condition for the theorem to hold, even more generally, in every residually free group.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128938693","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":"Two-party key establishment: From passive to active security without introducing new assumptions","authors":"Kashi Neupane","doi":"10.1515/gcc-2012-0005","DOIUrl":"https://doi.org/10.1515/gcc-2012-0005","url":null,"abstract":"Abstract. Key establishment protocols based on hardness assumptions, such as the discrete logarithm problem and the integer factorization problem, are vulnerable to quantum computer attacks, whereas the protocols based on other hardness assumptions, such as the conjugacy search problem and the decomposition search problem, can resist such attacks. The existing protocols based on the hardness assumptions which can resist quantum computer attacks are only passively secure. Compilers are used to convert a passively secure protocol to an actively secure protocol. Compilers involve some tools such as a signature scheme and a collision-resistant hash function. If there are only passively secure protocols but not a signature scheme based on the same assumption, then the application of existing compilers requires the use of such tools based on different assumptions. But the introduction of new tools, based on different assumptions, makes the new actively secure protocol rely on more than one hardness assumption. We offer an approach to derive an actively secure two-party protocol from a passively secure two-party protocol without introducing further hardness assumptions. This serves as a useful formal tool to transform any basic algebraic method of public key cryptography to the real world applicable cryptographic scheme.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127802815","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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