Algorithm for the fast generation of Latin squares

Hannusch, Carolin; Major, Roland; Pavone, Marco

doi:10.47974/JDMSC-2306

Open Access Research Article

Algorithm for the fast generation of Latin squares

^*Carolin HannuschCorresponding authorhannusch.carolin@inf.unideb.huFaculty of InformaticsKassai út 26University of DebrecenDebrecen, 4028, Hungary0000-0002-0098-7293View full profile → , Roland Majormajor.sandor@inf.unideb.huFaculty of InformaticsKassai út 26University of DebrecenDebrecen, 4028, Hungary0009-0000-3323-3327View full profile → , Marco Pavonemarco.pavone@unipa.itDipartimento di IngegneriaUniversità degli Studi di PalermoPalermo, 90128, Italy0000-0002-8674-5841View full profile →

* Corresponding author · click or hover a name for details

pp. 127–137Vol. 29Issue 1January 2026DOI: 10.47974/JDMSC-2306 Crossmark XML

Received:: 10 Sep 2024
Published Online:: 14 Aug 2025
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2306
Pages:: 127–137

Abstract

We introduce an algorithm for the fast generation of Latin squares. Considering Latin squares with n rows and columns, we introduce a map which maps one square to another one of opposite parity. This map is not unique and it is determined by two indices, i.e. two single numbers. These numbers can be used as secret key in cryptographic schemes which are build up on Latin squares. Thus the introduced algorithm can be applied in cryptography. Additionally, we give a theoretical upper bound for the number of Latin squares of order n.

Keywords

Generating discrete structures Number of Latin squares Cryptographic scheme Alon-Tarsi conjecture

Subject Classifications

05B1594A6065Y20

References

[1] G. Tarry, “Le probleme des 36 officiers,” Comptes rendus Ass. Fr. pour l’Avancement de Science Naturel, vol. 1, pp. 122–123 (1900).
[2] A. D. Keedwell and J. Dénes, Latin Squares and Their Applications. Amsterdam, Netherlands: North-Holland (2015).
[3] N. O. Schmidt, “Latin squares and their applications to cryptography,” M.Sc. thesis, Boise State University, Idaho, USA (2016).
[4] Y. Ren, F. Liu, T. Guo, R. Feng, and D. Lin, “Cheating prevention visual cryptography scheme using Latin square,” IET Information Security, vol. 11, no. 4, pp. 211–219 (2017).
[5] S. K. Pal, S. Kapoor, A. Arora, R. Chaudhary, and J. Khurana, “Design of strong cryptographic schemes based on Latin squares,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 13, no. 3, pp. 233–256 (2010).
[6] A. Desoky, H. Ammar, G. Fahmy, S. El-Sappagh, A. Hendawi, and S. H. Basha, “Latin square and artificial intelligence cryptography for blockchain and centralized systems,” in Proc. Int. Conf. on Advanced Intelligent Systems and Informatics, pp. 444–455 (2023).
[7] S. S. Kumar and S. A. Sherfin, “A cryptographic encryption technique byte – spiral rotation encryption algorithm,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 22, no. 3, pp. 371–376 (2019). https://doi.org/10.1080/09720529.2019.1578083.
[8] N. Jacobson, Basic Algebra I. New York, NY, USA: Courier Corporation (2012).
[9] R. Alter, “How many Latin squares are there?” The American Mathematical Monthly, vol. 82, no. 6, pp. 632–634 (1975).
[10] N. Alon and M. Tarsi, “Colorings and orientations of graphs,” Combinatorica, vol. 12, pp. 125–134 (1992).
[11] A. A. Drisko, “On the number of even and odd Latin squares of order p + 1,” Advances in Mathematics, vol. 128, no. 1, pp. 20–35 (1997).
[12] A. A. Drisko, “Proof of the Alon-Tarsi conjecture for n = 2^rp,” Electronic Journal of Combinatorics, vol. 5, no. 1, p. 28 (1998).

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