Parastrophically equivalent quasigroup equations

Krapež, Aleksandar; Živković, Dejan

doi:10.2298/pim1001039k

Cited by 7 publications

(15 citation statements)

References 5 publications

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“…6,8 It encrypts blocks of text by parastrophes of a single quasigroup and provably increases resistance to brute force attacks. 6,8 It encrypts blocks of text by parastrophes of a single quasigroup and provably increases resistance to brute force attacks.…”

Section: Block Ciphersmentioning

confidence: 99%

“…The transformations e (a,•) and d (a,⧵) are mutually inverse permutations of Q + for every leader a ∈ Q and every input sequence q ∈ Q + . Together with algebraic concepts such as isotopy 5,8,12,15,17,25,36,46 and parastrophy, 5,14,15,35,42,46 they have been used to establish a wide range of complex cryptographic schemes built on top of carefully selected quasigroups with cryptographically favorable properties. Together with algebraic concepts such as isotopy 5,8,12,15,17,25,36,46 and parastrophy, 5,14,15,35,42,46 they have been used to establish a wide range of complex cryptographic schemes built on top of carefully selected quasigroups with cryptographically favorable properties.…”

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

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: Block Ciphersmentioning

confidence: 99%

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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“…But this definition is very complicated. Krapež andŽivković proved in [13] that a generalized quadratic quasigroup functional equation Eq is parastrophically uncancellable iff the relation ∼ is the full relation on functional variables of Eq iff the relation ≡ is the full relation on vertices of K(Eq) iff K (Eq) is three-connected. We shall use this characterization of parastrophic uncancellability instead of the original definition.…”

Section: Lemma 212mentioning

confidence: 99%

“…Krapež andŽivković [13] solved this problem by reducing it to the problem of classification of three-connected finite cubic graphs. They refined the theory developed by Krstić in [14] which proves that there is a bijection between sets of parastrophically equivalent generalized quadratic quasigroup equations on one side and connected cubic graphs on the other.…”

Section: Introductionmentioning

confidence: 99%

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Parastrophically uncancellable quasigroup equations

2010

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Krstić initiated the use of cubic graphs in solving quasigroup equations. Based on his work, Krapež andŽivković proved that there is a bijective correspondence between classes of parastrophically equivalent parastrophically uncancellable generalized quadratic functional equations on quasigroups and three-connected cubic (multi)graphs. We use the list of such graphs given in the literature to verify existing results on equations with three, four and five variables and to prove new results for equations with six variables. We start with 14 nonisomorphic graphs with ten vertices, choose a set of 14 representative parastrophically nonequivalent equations and give their general solutions. A case of equations with seven and more variables is briefly discussed. The problem of Sokhats'kyi concerning a property which distinguishes visually two parastrophically nonequivalent equations with four variables is solved.Mathematics Subject Classification (2010). Primary 39B52; Secondary 20N05, 05C25.

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Cryptographically Suitable Quasigroups via Functional Equations

Krapež

2013

ICT Innovations 2012

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Parastrophically equivalent quasigroup equations

Cited by 7 publications

References 5 publications

An acceleration of quasigroup operations by residue arithmetic

An acceleration of quasigroup operations by residue arithmetic

Parastrophically uncancellable quasigroup equations

Cryptographically Suitable Quasigroups via Functional Equations

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