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Generation of n-quasigroups by proper families of functions

  • Aleksey V. Galatenko EMAIL logo , Valentin A. Nosov , Anton E. Pankratiev und Kirill D. Tsaregorodtsev
Veröffentlicht/Copyright: 6. September 2025
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Abstract

Finite quasigroups and n-quasigroups are a promising platform for cryptoalgorithm implementation. One of the key problems consists in memory-efficient generation of wide classes of n-quasigroups of a large order. We describe a possible solution based on proper families of functions, show that the number of n-quasigroups generated thereby is bounded from below in terms of the cardinality of the image of the corresponding proper family, study possible values that this cardinality can take, and give two examples of quadratic proper families of Boolean functions with a high image cardinality.


Originally published in Diskretnaya Matematika (2023) 35, №1, 35–53 (in Russian).


Funding statement: This research was supported by the Interdisciplinary Scientific and Educational School of Moscow University «Brain, Cognitive systems, Artificial intelligence».

References

[1] Glukhov M. M., “Some applications of quasigroups in cryptography”, Prikladnaya diskretnaya matematika, 2008, № 2(2), 28–32.10.17223/20710410/2/7Suche in Google Scholar

[2] Shcherbacov V. A., “Quasigroups in cryptology”, Computer Science J. Moldova, 172(50) (2009), 193–228.Suche in Google Scholar

[3] Chauhan D., Gupta I., Verma R., “Quasigroups and their applications in cryptography”, Cryptologia, 453 (2021), 227–265.10.1080/01611194.2020.1721615Suche in Google Scholar

[4] Markovski S., Mileva A., “NaSHA — family of cryptographic hash functions”, The First SHA-3 Candidate Conference (Leuven, Belgium), 2009.Suche in Google Scholar

[5] Gligoroski D., Ødegård R. S., Mihova M., Knapskog S. J., Drápal A., Klima V., Amundse J., El-Hadedy M., “Cryptographic hash function EDON-R’”, Proc. 1st Int. Workshop Security and Communic. Networks, IWSCN, 2009, 1–9.Suche in Google Scholar

[6] Gligoroski D., Mihajloska H., Otte D., El-Hadedy M., GAGE and InGAGE http://gageingage.org/upload/GAGEandInGAGEv1.03.pdf.Suche in Google Scholar

[7] Xu M., Tian Z., “An image cipher based on Latin cubes”, Proc. 3rd Int. Conf. Inf. Computer Technol., 2020, 160–168.10.1109/ICICT50521.2020.00033Suche in Google Scholar

[8] Dömösi P., Horváth G., “A novel cryptosystem based on abstract automata and Latin cubes”, Stud. Sci. Math. Hungar., 522 (2015), 221–232.10.1556/012.2015.52.2.1309Suche in Google Scholar

[9] Markovski S., Mileva A., “Generating huge quasigroups from small non-linear bijections via extended Feistel function”, Quasigroups and Relat. Syst., 171 (2009), 97–106.Suche in Google Scholar

[10] Nosov V. A., “The criterion of regularity of a Boolean nonautonomous automaton with split input”, Intellekt. sistemy, 3 1998, 269–280 (in Russain).Suche in Google Scholar

[11] Nosov V. A., “Construction of Latin square classes in a Boolean database”, Intellekt. sistemy, 4 1999, 307–320 (in Russain).Suche in Google Scholar

[12] Nosov V. A., Pankratiev A. E., “Latin squares over Abelian groups”, J. Math. Sci., 1493 (2008), 1230–1234.10.1007/s10958-008-0061-9Suche in Google Scholar

[13] Galatenko A. V., Nosov V. A., Pankratiev A. E., “Latin squares over quasigroups”, Lobachevskii J. Math., 412 (2020), 194–203.10.1134/S1995080220020079Suche in Google Scholar

[14] Plaksina I. A., “Construction of a parametric family of multidimensional Latin squares”, Intellekt. sistemy, 18 2014, 323–329 (in Russain).Suche in Google Scholar

[15] Chen Y., Gligoroski D., Knapskog S., “On a special class of multivariate quadratic quasigroups (MQQs)”, J. Math. Cryptology, 78 (2013), 111–144.10.1515/jmc-2012-0006Suche in Google Scholar

[16] Lau D., Function Algebras on Finite Sets: A Basic Course on Many-valued Logic and Clone Theory, Springer, 2006, 684 pp.Suche in Google Scholar

[17] Galatenko A. V., Nosov V. A., Pankratiev A. E., “Generation of Multivariate Quadratic Quasigroups by Proper Families of Boolean Functions”, J. Math. Sci., 262 (2022), 630–641.10.1007/s10958-022-05843-7Suche in Google Scholar

[18] Tsaregorodtsev K. D., “Properties of proper families of Boolean functions”, Discrete Mathematics and Applications, 325 (2022), 369–378.10.1515/dma-2022-0030Suche in Google Scholar

[19] Krotov D. S., Potapov V. N., Sokolova P. V., “On reconstructing reducible n-ary quasigroups and switching subquasigroups”, Quasigroups and Relat. Syst., 161 (2008), 55–67.Suche in Google Scholar

[20] Tsaregorodtsev K. D., “One-to-one correspondense between proper families of Boolean functions and unique sink orientations of cube”, Prikladnaya diskretnaya matematika, 2020, № 48, 16–21 (in Russian).10.17223/20710410/48/2Suche in Google Scholar

[21] Vajda S., Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, E. Horwood Ltd, 1989, 189 pp.Suche in Google Scholar

[22] Zuev Yu. A., Across the ocean of discrete mathematics. V. 1, URSS, 2012 (in Russian), 274 pp.Suche in Google Scholar

[23] Galatenko A. V., Pankratiev A. E., Staroverov V. M., “Generation of proper families of functions”, Lobachevskii J. Math., 433 (2022), 571–581.10.1134/S1995080222060105Suche in Google Scholar

Received: 2022-11-28
Published Online: 2025-09-06
Published in Print: 2025-08-26

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 30.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/dma-2025-0015/pdf
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