On congruences of groupoids closely connected with quasigroups

Shcherbacov, Victor; Tabarov, A. Kh.; Puşcaşu, D. I.

doi:10.1007/s10958-009-9716-4

Cited by 10 publications

(16 citation statements)

References 10 publications

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“…Following this logic we can denote identity ( 5) by (SP) and identity ( 6) by (IP) since these identities guarantee that middle translations (P) are respectively surjective and injective mappings relative to the operation "•" [19]. Indeed, using Table 1, we see that L / x = P −1…”

Section: Quasigroupmentioning

confidence: 99%

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Units in generalized derivatives of quasigroups

Horosh,

Malyutina,

Scerbacova

et al. 2020

Preprint

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We proceed the research of generalized quasigroup derivatives started in early papers of the last co-author ([20, p. 212], [13]). For any quasigroup there exist 648 generalized derivatives. Here we study the problem about existence of units (left, right, middle) in quasigroups that are a generalized derivative of a quasigroup. There exist 1944 various cases. For every case we find a proof or a counterexample, often using Prover or Mace [15,14].

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Section: Quasigroupmentioning

confidence: 99%

“…Lemma 2. In algebra (Q, •, \, /) with identities ( 1)-( 4), identities ( 5) and ( 6) are true [21,18,19].…”

Section: Quasigroupmentioning

confidence: 99%

Units in generalized derivatives of quasigroups

Horosh,

Malyutina,

Scerbacova

et al. 2020

Preprint

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“…59 The left parastrophe of a quasigroups is another quasigroup, (Q, ⧵), with the same carrier set, Q, and a multiplication, ⧵, that satisfies 2 59 The left parastrophe of a quasigroups is another quasigroup, (Q, ⧵), with the same carrier set, Q, and a multiplication, ⧵, that satisfies 2…”

Section: Definition 3 (A Simple Quasigroup Stream Cipher)mentioning

confidence: 99%

An acceleration of quasigroup operations by residue arithmetic

Krömer

Platoš

Nowaková

et al. 2017

Concurrency and Computation

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Quasigroup operations are essential for a wide range of cryptographic procedures that includes cryptographic hash functions, electronic signatures, pseudorandom number generators, and stream and block ciphers. Quasigroup cryptography achieves high levels of security at low memory and computational costs by an iterative application of quasigroup operations to streams and blocks of data. The use of large quasigroups can further improve the strength of cryptographic operations. However, the order of used quasigroups is the main factor affecting the memory requirements of quasigroup cryptographic schemes. Alternative quasigroup representations that do not store their multiplication tables in computer memory yield increased computational costs.In any case, an efficient implementation of quasigroup operations is critical for practical applications of quasigroup cryptography. Residue number systems allow a fast, concurrent realization of addition and multiplication. In this work, residue arithmetic is used to accelerate quasigroup operations, and an efficient computational approach to their implementation, designed with respect to the extended instruction sets of modern processors, is proposed. KEYWORDSacceleration, quasigroup operations, residue arithmetic, residue number systems, security INTRODUCTIONQuasigroups are algebraic structures with a large number of applications in cryptography and computer security. 1-5 A quasigroup consists of a set of elements (carrier) and a binary operation (multiplication) that maps every 2 of its elements onto a third element. Mathematically, they are grupoids closely related to the combinatorial concept of latin squares 2,3,6,7 and the geometric notion of k−nets (3−nets). 2,8 Multiplication (Cayley) tables of finite quasigroups can be expressed as latin squares, and k−nets can be coordinatized by systems of orthogonal quasigroups.In general, quasigroups are noncommutative, nonassociative, nonidempotent, and noninvolutory and do not have neutral elements. 9 These properties make them suitable for applications in cryptography, and they have been used as fundamental elements of a wide range of cryptographic algorithms with success. Their applications in this area include novel high-performance stream 4-6,10-13 and block 4,5,14-22 ciphers, cryptographic hash 4,5,23-27 and trapdoor 4,14,23 functions, fast and secure electronic signatures (eg, message authentication codes), 4,14,23,28 pseudorandom number generators, 5,29-31 error-detecting 2 and error-correcting 2,9,32 codes, and simultaneous encryption and error-correction coding (cryptcoding). 4Quasigroups can be also used to model cryptosystems based on other cryptographic primitives. A quasigroup representation has been devised for block ciphers based on Feistel networks (ie, Feistel and generalized Feistel ciphers). 33The attractiveness of quasigroups for practical cryptographic applications consists in the combination of provably strong security and inexpensive realization of quasigroup-based cryptographic operations. They support la...

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“…Connections between different kinds of local identity elements in different parastrophes of a quasigroup (Q, ·) are given in the following table [93,94]. Table 2 ε = · (12) = * (13) = / (23) = \ (123) = // (132)…”

Section: Lemma 11 In a Quasigroup (Q A): (At )mentioning

confidence: 99%

“…Let (Q, +) be an IP-loop, x · y = (ϕx + ψy) + c, where ϕ, ψ ∈ Aut(Q, +), a ∈ C(Q, +), θ be a normal congruence of (Q, +). Then θ is normal congruence of (Q, ·) if and only if ϕ | Ker θ , ψ | Ker θ are automorphisms of Ker θ [80,59,94,96].…”

Section: Theorem 110 a Subquasigroup H Of A Quasigroup Q Is Normal mentioning

confidence: 99%

On the structure of left and right F-, SM-, and E-quasigroups

Shcherbacov¹

2008

J Generalized Lie Theory Appl

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Abstract. It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM-and E-quasigroups.Information on simple quasigroups from these quasigroup classes is given, for example, finite simple Fquasigroup is a simple group or a simple medial quasigroup.It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isomorphic the direct product of a group and a left S-loop (this is an answer to Belousov "1a" problem).Any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.2000 Mathematics Subject Classification: 20N05

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On congruences of groupoids closely connected with quasigroups

Cited by 10 publications

References 10 publications

Units in generalized derivatives of quasigroups

Units in generalized derivatives of quasigroups

An acceleration of quasigroup operations by residue arithmetic

On the structure of left and right F-, SM-, and E-quasigroups

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