Abstract
We investigate sets of mutually orthogonal latin squares (MOLS) generated by cellular automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order \(q^{d-1}\), we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field \(\mathbb {F}_q\) are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree n over \(\mathbb {F}_q\). Finally, we present a construction for families of MOLS based on LBCA, and prove that their cardinality corresponds to the maximum number of pairwise coprime polynomials with nonzero constant term. Although our construction does not yield all such families of MOLS, we show that the resulting lower bound is asymptotically close to their actual number.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allender E., Bernasconi A., Damm C., von zur Gathen J., Saks M.E., Shparlinski I.E.: Complexity of some arithmetic problems for binary polynomials. Comput. Complex. 12((1–2)), 23–47 (2003).
Benjamin A.T., Bennett C.D.: The probability of relatively prime polynomials. Math. Mag. 80(3), 196–202 (2007).
Colbourn C.J.: Construction techniques for mutually orthogonal latin squares. In: Combinatorics Advances, pp. 27–48. Springer, Berlin (1995).
del Rey Á.M., Mateus J.P., Sánchez G.R.: A secret sharing scheme based on cellular automata. Appl. Math. Comput. 170(2), 1356–1364 (2005).
Deißler J.: A resultant for Hensel’s lemma. arXiv preprint arXiv:1301.4073 (2013).
Eloranta K.: Partially permutive cellular automata. Nonlinearity 6(6), 1009–1023 (1993).
Gauß C.F.: Disquisitiones arithmeticae. Humboldt-Universität zu Berlin (1801).
Gelfand I.M., Kapranov M., Zelevinsky A.: Discriminants, Resultants, and Multidimensional Determinants. Springer, Berlin (2008).
Golomb S.W., Posner E.C.: Rook domains, latin squares, affine planes, and error-distributing codes. IEEE Trans. Inf. Theory 10(3), 196–208 (1964).
Gorodilova A., Agievich S., Carlet C., Hou X., Idrisova V., Kolomeec N., Kutsenko A., Mariot L., Oblaukhov A., Picek S., Preneel B., Rosie R., Tokareva N.N.: The Fifth International Students’ Olympiad in Cryptography—NSUCRYPTO: Problems and their Solutions. CoRR abs/1906.04480 (2019).
Hedlund G.A.: Endomorphisms and automorphisms of the shift dynamical systems. Math. Syst. Theory 3(4), 320–375 (1969).
Hou X., Mullen G.L.: Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields. Finite Fields Appl. 15(3), 304–331 (2009).
Kari J.: Theory of cellular automata: a survey. Theor. Comput. Sci. 334(1–3), 3–33 (2005).
Keedwell A.D., Dénes J.: Latin Squares and their Applications. Elsevier, Amsterdam (2015).
Lidl R., Niederreiter H.: Introduction to Finite Fields and their Applications. Cambridge University Press, Cambridge (1994).
MacNeish H.F.: Euler squares. Ann. Math. 23, 221–227 (1922).
Mariot L., Formenti E., Leporati A.: Constructing orthogonal latin squares from linear cellular automata. CoRR abs/1610.00139. http://arxiv.org/abs/1610.00139 (2016).
Mariot L., Formenti E., Leporati A.: Enumerating orthogonal latin squares generated by bipermutive cellular automata. In: Proceedings of the Cellular Automata and Discrete Complex Systems—23rd IFIP WG 1.5 International Workshop, AUTOMATA 2017, Milan, Italy, 7–9 June 2017, pp. 151–164 (2017).
Mariot L., Leporati A.: Sharing secrets by computing preimages of bipermutive cellular automata. In: Proceedings of the Cellular Automata—11th International Conference on Cellular Automata for Research and Industry, ACRI 2014, Krakow, Poland, 22–25 Sept 2014, pp. 417–426 (2014).
Mariot L., Leporati A.: A cryptographic and coding-theoretic perspective on the global rules of cellular automata. Nat. Comput. 17(3), 487–498 (2018).
Mariot L., Leporati A., Dennunzio A., Formenti E.: Computing the periods of preimages in surjective cellular automata. Nat. Comput. 16(3), 367–381 (2017).
Mariot L., Picek S., Leporati A., Jakobovic D.: Cellular automata based S-boxes. Cryptogr. Commun. 11(1), 41–62 (2019).
Montgomery D.C.: Design and Analysis of Experiments. Wiley, Hoboken (2017).
Moore C.: Predicting nonlinear cellular automata quickly by decomposing them into linear ones. Phys. D: Nonlinear Phenom. 111(1–4), 27–41 (1998).
Moore C., Drisko A.A., et al.: Algebraic properties of the block transformation on cellular automata. Complex Syst. 10(3), 185–194 (1996).
Pedersen J.: Cellular automata as algebraic systems. Complex Syst. 6(3), 237–250 (1992).
Reifegerste A.: On an involution concerning pairs of polynomials over \({\mathbb{F}}_2\). J. Comb. Theory Ser. A 90(1), 216–220 (2000).
Stinson D.R.: Combinatorial characterizations of authentication codes. Des. Codes Cryptogr. 2(2), 175–187 (1992).
The Online Encyclopedia of Integer Sequences (OEIS). Sequence A002450. http://oeis.org/A002450. Accessed 12 Apr 2019
Vaudenay S.: On the need for multipermutations: Cryptanalysis of MD4 and SAFER. In: Proceedings of the Fast Software Encryption: Second International Workshop, Leuven, Belgium, 14–16 Dec 1994, pp. 286–297 (1994).
Wilson R.M.: Concerning the number of mutually orthogonal latin squares. Discret. Math. 9(2), 181–198 (1974).
Acknowledgements
The authors wish to thank Arthur Benjamin, Curtis Bennett and Igor Shparlinski for their insightful suggestions on how to count the number of pairs of coprime polynomials with nonzero constant term. Further, the authors thank the anonymous reviewers for their useful comments to improve the readability of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. J. Colbourn.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work has been partially supported by COST Action IC1405, “Reversible Computation—Extending the Horizons of Computing”.
Rights and permissions
About this article
Cite this article
Mariot, L., Gadouleau, M., Formenti, E. et al. Mutually orthogonal latin squares based on cellular automata. Des. Codes Cryptogr. 88, 391–411 (2020). https://doi.org/10.1007/s10623-019-00689-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-019-00689-8