zbMATH — the first resource for mathematics

Latin squares over quasigroups. (English) Zbl 1451.05030
In this paper, the authors describe a constructive way for generating finite quasigroups of any given order that generalizes a previous method defined in [V. A. Nosov and A. E. Pankratiev, J. Math. Sci., New York 149, No. 3, 1230–1234 (2008; Zbl 1146.05011); translation from Fundam. Prikl. Mat. 12, No. 3, 65–71 (2006)]. Unlike that work, where they made use of proper families of functions over abelian groups, the new construction is based on proper families of functions over quasigroups and 3-quasigroups. In addition, unlike the previous proposal, the new one enables one to generate finite quasigroups containing no subquasigroups.
05B15 Orthogonal arrays, Latin squares, Room squares
20N05 Loops, quasigroups
Full Text: DOI
[1] Shannon, C., Communication theory of secrecy systems, Bell Syst. Tech. J., 28, 656-715 (1949) · Zbl 1200.94005
[2] M. M. Glukhov, ‘‘On appplications of quasigroups in cryptography,’’ Appl. Discrete Math., No. 2, 28-32 (2008). · Zbl 1448.94198
[3] Artamonov, V. A.; Chakrabarti, S.; Gangopadhyay, S.; Pal, S. K., On Latin squares of polynomially complete quasigroups and quasigroups generated by shifts, Quasigroups Rel. Syst., 21, 117-130 (2013) · Zbl 1294.20073
[4] Artamonov, V. A.; Chakrabarti, S.; Pal, S. K., Characterizations of highly non-associative quasigroups and associative triples, Quasigroups Rel. Syst., 25, 1-19 (2017) · Zbl 1373.20082
[5] Galatenko, A. V.; Pankratiev, A. E.; Rodin, S. B., Polynomially complete quasigroups of prime order, Algebra Logic, 57, 327-335 (2018) · Zbl 1414.20023
[6] Galatenko, A. V.; Pankratiev, A. E., The complexity of decision of polynomial completeness of finite quasigroups, Discrete Math., 30, 3-11 (2018)
[7] Nosov, V. A., Construction of classes of latin squares in a boolean database, Intell. Syst., 4, 307-320 (1999)
[8] Nosov, V. A.; Pankratiev, A. E., Latin squares over Abelian groups, J. Math. Sci., 149, 1230-1234 (2008) · Zbl 1146.05011
[9] Nosov, V. A., Construction of a parametric family of Latin squares in a vector database, Intell. Syst., 8, 517-528 (2004)
[10] Kepka, T., “A note on simple quasigroups,” Acta Univ. Carolinae, Math. Phys., 19, 59-60 (1978) · Zbl 0383.20058
[11] V. A. Nosov and A. E. Pankratiev, ‘‘A generalization of the Feistel cipher,’’ in Proceedings of the International Conference Maltsev Readings, Novosibirsk, 2015, p. 59.
[12] Piven, N. A., Investigation of quasigroups generated by proper families of Boolean functions of order 2, Intell. Syst. Theory Appl., 22, 21-35 (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.