Isotopy of latin squares in cryptography

Grošek, Otokar; Sýs, Marek

doi:10.2478/v10127-010-0003-z

Cited by 8 publications

(9 citation statements)

References 9 publications

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“…-Algorithmics: isomorphism testing (see Babai and Qiao (2012); Grošek and Sýs (2010); Huber (2011);Miller (1978); Rosenbaum (2012)), autocorrelations of strings (see Guibas and Odlyzko (1981); Rivals and Rahmann (2003)), information theory (see Abu-Mostafa (1986)), random digital search trees (see Drmota (2009)), population recovery (see Wigderson and Yehudayoff (2012)), and asymptotics of recurrences (see Knuth (1966); O' Shea (2004));…”

Section: Discussionmentioning

confidence: 99%

Analysis of an Exhaustive Search Algorithm in Random Graphs and the $n^{c\log n}$-Asymptotics

Banderier¹,

Hwang²,

Ravelomanana³

et al. 2014

SIAM J. Discrete Math.

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We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual G n,p -model where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n c log n , which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms.MSC 2000 Subject Classifications: Primary 05C80, 05C85; secondary 65Q30.

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Section: Discussionmentioning

confidence: 99%

Analysis of an Exhaustive Search Algorithm in Random Graphs and the $n^{c\log n}$-Asymptotics

Banderier¹,

Hwang²,

Ravelomanana³

et al. 2014

SIAM J. Discrete Math.

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“…New results on isotopisms [134], autotopisms [135], automorphisms [136] and parastrophisms [137] of quasigroups have continued to progress until the present day. Furthermore, different applications of autotopisms of quasigroups in Cryptography have been developed [138,139].…”

Section: Quasigroups Latin Squares and Related Structuresmentioning

confidence: 99%

A Historical Perspective of the Theory of Isotopisms

2018

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In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.

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“…Non-associative structures, in particular quasigroups seem to have a long history in cryptography. For an overview on cryptographic applications of quasigroups and Latin squares, see [Shc09,GS10,Shc12]. In particular, we mention the work of Denes and Keedwell [DK74, DK91, DK92, DK02].…”

Section: Introductionmentioning

confidence: 99%

Non-associative public-key cryptography

Kalka¹

2016

Algebra and Computer Science

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We introduce a generalized Anshel-Anshel-Goldfeld (AAG) key establishment protocol (KEP) for magmas. This leads to the foundation of non-associative public-key cryptography (PKC), generalizing the concept of non-commutative PKC. We show that left selfdistributive systems appear in a natural special case of a generalized AAG-KEP for magmas, and we propose, among others instances, concrete realizations using fconjugacy in groups and shifted conjugacy in braid groups. We discuss the advantages of our schemes compared with the classical AAG-KEP based on conjugacy in braid groups.KEP's based on the shifted and f -conjugacy problem. In section 5 we discuss generalizations, like AAG-schemes over non-associative magmas, open problems and further work.Summary. The main purpose of this paper is popularize the notion of nonassociative cryptography and to provide a general framework for non-associative and non-commutative KEP's by utilizing the unifying approach that stems from the general AAG-KEP for magmas. We argue for the superiority of the non-associative schemes introduced in section 4 compared to classical noncommutative AAG commutator KEP. Anyway, in our opinion the field of non-commutative cryptography lacked over the last years supply of new innovative cryptosystems. We hope that nonassociative cryptography will contribute to revived interest in non-commutative cryptography.Outlook. Nevertheless, this is not the end, rather the beginning of the story of non-associative cryptography.In the forthcoming paper [KaT12], by introducing a small asymmetry in the nonassociative AAG protocol for magmas, we succeed to construct non-associative KEP's for all LD-and multi-LD-systems (in general: sets with distributive operations). We consider the systems and instances given in [KaT12] as much more practical and interesting than the one given in this paper. In particular, since these systems work for all LD-and multi-LD-systems, they deploy two further advantages. First, we may consider encryption functions using iterated multiplication (in the magma) from the left. Therefore, in order to obtain the secret key an attacker has to solve an iterated f -or shifted conjugacy problem. Second, for a given (partial) multi-LD-system it turns out that even the used operations can be hidden, i.e., they are part of the secret key.Historical remarks. Non-associative structures, in particular quasigroups seem to have a long history in cryptography. For an overview on cryptographic applications of quasigroups and Latin squares, see [Shc09, GS10, Shc12]. In particular, we mention the work of Denes and Keedwell [DK74, DK91, DK92, DK02]. Nevertheless, except for authentication schemes and zero-knowledge protocols, most of these applications are in classical (i.e. symmetric key) cryptography. The earliest quasigroup-based public-key cryptosystem that we are aware of is due to Koscielny and Mullen [KM99]. Non-associative cryptography that goes beyond quasigroups, in particular, the generalized AAG-KEP for magmas were introduced by the author in his...

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Isotopy of latin squares in cryptography

Cited by 8 publications

References 9 publications

Analysis of an Exhaustive Search Algorithm in Random Graphs and the $n^{c\log n}$-Asymptotics

Analysis of an Exhaustive Search Algorithm in Random Graphs and the $n^{c\log n}$-Asymptotics

A Historical Perspective of the Theory of Isotopisms

Non-associative public-key cryptography

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