Polynomially complete quasigroups of prime order

Galatenko, A. V.; Pankratiev, A. E.; Rodin, S. B.

doi:10.33048/alglog.2018.57.501

Cited by 2 publications

(1 citation statement)

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“…However, its intensive use by the crypto community can be traced back only in the last two decades, see Belyavskaya (1994); Shcherbacov (2017); Smith (2007). The significant properties of quasigroups for the crypto designs are polynomial completeness (simplicity, non-affineness), degree of non-commutativity and non-associativity, nonexistence of proper subquasigroup, see Artamonov (2012); Artamonov et al (2013Artamonov et al ( , 2017; Galatentko et al (2018); Grošek and Horák (2012).…”

Section: Introductionmentioning

confidence: 99%

Algebraic Properties of Subquasigroups and Construction of Cryptographically Suitable Finite Quasigroups

Артамонов¹,

Chakrabarti²,

Tiwari³

et al. 2021

Preprint

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In this paper, we identify many important properties and develop criteria for the existence of subquasigroups in finite quasigroups. Based on these results, we propose an effective method that concludes the nonexistence of subquasigroup of a finite quasigroup, otherwise finds its all possible proper subquasigroups. This has an important application in checking the cryptographic suitability of a quasigroup.Further, we propose a binary operation using arithmetic of finite fields to construct quasigroups of order p r . We develop the criteria under which these quasigroups have desirable cryptographic properties, viz. polynomially completeness and possessing no proper subquasigroups. Then a practical method is given to construct cryptographically suitable quasigroups. We also illustrate these methods by some academic examples and implement all proposed algorithms in the computer algebra system SINGULAR.

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Section: Introductionmentioning

confidence: 99%