Quasigroups in cryptology

Shcherbacov, Victor

doi:10.1201/9781315120058-14

Cited by 9 publications

(12 citation statements)

References 45 publications

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“…The decryption procedure is realized by a d−transformation defined on a (23)−parastrophe (left parastrophe) of (Q, •). 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%

“…It can be applied to encrypt an input stream of elements by a series of quasigroup multiplications. The decryption procedure is realized by a d −transformation defined on a (23)−parastrophe (left parastrophe) of ( Q ,∘) . The left parastrophe of a quasigroups is another quasigroup, ( Q ,∖), with the same carrier set, Q , and a multiplication, ∖, that satisfies

\forall false(u, v \in Q false) false(u ∖ false(u \circ v false) = v \land u \circ false(u ∖ v false) = v false) .

The multiplication is sometimes denoted with reference to the original operation as ∘ (23) or ∖ ∘ .…”

Section: Elementary Constructs Of Quasigroup Cryptographymentioning

confidence: 99%

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

confidence: 99%

\forall false(u, v \in Q false) false(u ∖ false(u \circ v false) = v \land u \circ false(u ∖ v false) = v false) .

The multiplication is sometimes denoted with reference to the original operation as ∘ (23) or ∖ ∘ .…”

Section: Elementary Constructs Of Quasigroup Cryptographymentioning

confidence: 99%

An acceleration of quasigroup operations by residue arithmetic

Krömer

Platoš

Nowaková

et al. 2017

Concurrency and Computation

View full text Add to dashboard Cite

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“…Applications of quasigroups in cryptography are well documented in literature (see [37,38,40,48]). There are many broken designs based on quasigroups, but there are some with perfect crypto properties [40].…”

Section: Introductionmentioning

confidence: 99%

“…The most desirable quasigroups for building crypto primitives are the class of shapeless quasigroups. For details on shapelessness of quasigroup the reader can check [40,48]. As reported in [49], Moskovich expressed an interesting statement on his blog that "while associativity caters to the classical world of space and time, distributivity is perhaps the setting for the emerging world of information".…”

Section: Introductionmentioning

confidence: 99%

“…No wonder the African Agenda 2063 and the Global Agenda 2030 and its sustainable Development Goal (SDG) have cybersecurity as their major focus among the 7 aspirations, 20 goals and 39 priority areas, targets and indicators [41]. Cryptology has been highly applauded in literature for being capable of tackling cyberinsecurity [37,38,48]. Cryptology encompasses both cryptography (ciphering) and cryptanalysis (deciphering) and looks at the mathematical problems that underlie them.…”

Section: Introductionmentioning

confidence: 99%

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Latin quandles and applications to cryptography

Jaiyeola¹,

Adeniran²,

Isere³

2021

Math. Appl.

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This work investigated some properties of Latin quandles that are applicable in cryptography. Four distinct cores of an Osborn loop (non-diassociative and non-power associative) were introduced and investigated. The necessary and sufficient conditions for these cores to be (i) (left) quandles (ii) involutory quandles (iii) quasi-Latin quandles and (iv) involutory quasi-Latin quandles were established. These conditions were judiciously used to build cipher algorithms for cryptography in some peculiar circumstances.

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Hardness of learning loops, monoids, and semirings

Chang

2014

Discrete Applied Mathematics

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Quasigroups in cryptology

Abstract: We give a review of some known published applications of quasigroups in cryptology.

Cited by 9 publications

References 45 publications

An acceleration of quasigroup operations by residue arithmetic

An acceleration of quasigroup operations by residue arithmetic

Latin quandles and applications to cryptography

Hardness of learning loops, monoids, and semirings

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