Identities with permutations leading to linearity of quasigroups

Belyavskaya, G. B.; Tabarov, A. Kh.

doi:10.1515/dma.2009.010

Cited by 5 publications

(7 citation statements)

References 8 publications

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“…Note that in Theorem 3.1, Theorem 3.1a, Theorem 4.1, and Theorem 4.2 we improve and extend the results of the Theorem 3 in [14].…”

Section: Remark 42supporting

confidence: 61%

“…In this paper we continue the study of quasigroups isotopic to groups and Abelian groups which was initiated in the works [14,15]. With the use of the class of identities with permutations involving three variables, which has been considered in the paper [14], we refine and significantly improve Theorem 3 from [14], which provides a sufficiently general characteristic of quasigroups isotopic to groups and Abelian groups.…”

Section: Introductionmentioning

confidence: 93%

“…With the use of the class of identities with permutations involving three variables, which has been considered in the paper [14], we refine and significantly improve Theorem 3 from [14], which provides a sufficiently general characteristic of quasigroups isotopic to groups and Abelian groups. We introduce the notion of the type of identity with permutations and present identities with permutations of various types characterizing quasigroups isotopic to Abelian groups (Theorem 2.1).…”

Section: Introductionmentioning

confidence: 99%

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Identities with permutations and quasigroups isotopic to groups and to Abelian groups

Belyavskaya¹

2013

Discrete Mathematics and Applications

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This paper continues the study of quasigroups isotopic to groups and to Abelian groups. For identities with permutations of a certain kind the notion of the type is introduced; identities with permutations of various types are presented, which characterize quasigroups isotopic to Abelian groups. The approach being used had allowed to establish a number of new unbalanced identities involving four variables in an equasigroup, as well as a number of identities involving three variables in an equasigroup with one nullary operation, each of which characterises a quasigroup isotopic to a group or to an Abelian group.

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“…Note that in Theorem 3.1, Theorem 3.1a, Theorem 4.1, and Theorem 4.2 we improve and extend the results of the Theorem 3 in [14].…”

Section: Remark 42supporting

confidence: 61%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Identities with permutations and quasigroups isotopic to groups and to Abelian groups

Belyavskaya¹

2013

Discrete Mathematics and Applications

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“…Such quasigroups form a class of medial quasigroups [2]. If ( Ω, • ) is an Abelian semigroup, then we will call (Ω, * ) a medial groupoid.…”

Section: Algorithm 1 Of Key Exchangementioning

confidence: 99%

Key agreement schemes based on linear groupoids

Барышников¹,

Baryshnikov²,

Катышев³

et al. 2017

Матем. вопр. криптогр.

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Abstract. Authors present a study of the possibility to use special class of nonassociative groupoids (called linear) for the implementation of a key exchange protocol based on a generalization of Diffie -Hellmann algorithm. The necessity to use the power commutation and effective power calculation properties is proved. A specific example of linear groupoid over the elliptic curve is described.Keywords: key exchange protocol, Diffie -Hellmann algorithm, non-associative groupoids, linear quasigroups, elliptic curves Схемы выработки общего ключа на основе линейных группоидов А. В. Барышников, С. Ю. Катышев ООО «Центр сертификационных исследований», МоскваАннотация. Исследуется возможность использования одного класса неассо-циативных группоидов (так называемых линейных группоидов) для реализа-ции схемы выработки общего ключа, основанной на обобщении алгоритма Диффи -Хеллмана. Доказана необходимость использования коммутативности степеней и свойств эффективного вычисления степеней. Описан конкретный пример линейного группоида над группой точек эллиптической кривой.Ключевые слова: протокол выработки общего ключа, алгоритм Диффи -Хел-лмана, неассоциативные группоиды, линейные квазигруппы, эллиптические кривые.

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“…Such quasigroups form a class of medial quasigroups [3]. Analogously, if ( ,⋅) is an abelian semigroup we will call ( , * ) a medial groupoid.…”

Section: Locally Medial Groupoidsmentioning

confidence: 99%

Application of non-associative groupoids to the realization of an open key distribution procedure

Katyshev

Markov

Nechaev³

2015

Discrete Mathematics and Applications

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We investigate the possibility to use non-associative groupoids in the realization of an open key distribution procedure based on a generalization of the well known Di e-Hellman algorithm. We prove the existence of non-associative groupoids which are simultaneously power commuting and not power-associative. Note: Originally published in Diskretnaya Matematika (2014) 26, №3, 45-64 (in Russian).We study the possibility to generalize the well known Di e-Hellman algorithm [7] realizing an open key distribution protocol by means of cyclic group to the case when a non-associative groupoid is used instead of a group. Let us introduce necessary de nitions.For an element of a nite groupoid ( , * ) and given , ∈ ℕ we de ne the right -th and left -th powers respectively by equalitiesWe say that has commuting right powers, or that is a CRP-element ifIf this identity is valid for any element ∈ , then we say that ( , * ) is a CRP-groupoid. Similarly, by the identity ∀ , ∈ ℕ : [ ][ ] = [ ][ ] , we de ne elements and groupoids with commuting left powers, CLP-elements and CLP-groupoids, respectively. Algorithm 1 of the open key distribution. After choosing a (non-secret) CRP-element of a groupoid the users and independently choose arbitrary numbers , ∈ ℕ and exchange the elements [ ] and [ ] with each other. Then they form the common secret key [ ][ ] = [ ][ ] . The complexity of revealing the secret key by an observer which has access to the open information , [ ] , [ ] does not exceed the complexity of the right discrete logarithm in the groupoid computation, i.e. the complexity of solving the equation [ ] = ℎ.A natural generalization of this approach to the construction of open key distribution procedures is the combination of right and left powers.We say that a groupoid ( , * ) is a groupoid with commuting powers (CP-groupoid) if it is a CLP-and CRPgroupoid and, moreover, for any ∈ and any , ∈ ℕ the following equality holds:We say that an element of any groupoid ( , * ) is a CP-element if it generates a CP-groupoid.Let be a CP-element.

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Identities with permutations leading to linearity of quasigroups

Cited by 5 publications

References 8 publications

Identities with permutations and quasigroups isotopic to groups and to Abelian groups

Identities with permutations and quasigroups isotopic to groups and to Abelian groups

Key agreement schemes based on linear groupoids

Application of non-associative groupoids to the realization of an open key distribution procedure

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