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Inversion of Mutually Orthogonal Cellular Automata

  • Luca MariotEmail author
  • Alberto Leporati
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11115)

Abstract

Mutually Orthogonal Cellular Automata (MOCA) are sets of bipermutive CA which can be used to construct pairwise orthogonal Latin squares. In this work, we consider the inversion problem of pairs of configurations in MOCA. In particular, we design an algorithm based on coupled de Bruijn graphs which solves this problem for generic MOCA, without assuming any linearity on the underlying bipermutive rules. Next, we analyze the computational complexity of this algorithm, remarking that it runs in exponential time with respect to the diameter of the CA rule, but that it can be straightforwardly parallelized to yield a linear time complexity. As a cryptographic application of this algorithm, we finally show how to design a (2, n) threshold Secret Sharing Scheme (SSS) based on MOCA where any combination of two players can reconstruct the secret by applying our inversion algorithm.

Keywords

Cellular automata Latin squares Secret sharing schemes de Bruijn graph 

References

  1. 1.
    Benjamin, A.T., Bennett, C.D.: The probability of relatively prime polynomials. Math. Mag. 80(3), 196–202 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Betel, H., de Oliveira, P.P.B., Flocchini, P.: Solving the parity problem in one-dimensional cellular automata. Nat. Comput. 12(3), 323–337 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, 1st edn, pp. 257–397. Cambridge University Press, New York. (2010)Google Scholar
  4. 4.
    Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical systems. Math. Syst. Theory 3(4), 320–375 (1969)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Keedwell, A.D., Dénes, J.: Latin Squares and Their Applications. Elsevier, New York City (2015)zbMATHGoogle Scholar
  6. 6.
    Mariot, L.: Cellular automata, Boolean functions and combinatorial designs (2018). https://boa.unimib.it/bitstream/10281/199011/2/phd_unimib_701962.pdf
  7. 7.
    Mariot, L., Formenti, E., Leporati, A.: Constructing orthogonal Latin squares from linear cellular automata. CoRR abs/1610.00139 (2016)Google Scholar
  8. 8.
    Mariot, L., Formenti, E., Leporati, A.: Enumerating orthogonal Latin squares generated by bipermutive cellular automata. In: Dennunzio, A., Formenti, E., Manzoni, L., Porreca, A.E. (eds.) AUTOMATA 2017. LNCS, vol. 10248, pp. 151–164. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-58631-1_12CrossRefzbMATHGoogle Scholar
  9. 9.
    Mariot, L., Leporati, A.: Sharing secrets by computing preimages of bipermutive cellular automata. In: Wąs, J., Sirakoulis, G.C., Bandini, S. (eds.) ACRI 2014. LNCS, vol. 8751, pp. 417–426. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-11520-7_43CrossRefzbMATHGoogle Scholar
  10. 10.
    Mariot, L., Leporati, A., Dennunzio, A., Formenti, E.: Computing the periods of preimages in surjective cellular automata. Nat. Comput. 16(3), 367–381 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mariot, L., Picek, S., Jakobovic, D., Leporati, A.: Evolutionary algorithms for the design of orthogonal Latin squares based on cellular automata. In: Proceedings of the Genetic and Evolutionary Computation Conference. GECCO 2017, 15–19 July 2017, Berlin, Germany, pp. 306–313 (2017)Google Scholar
  12. 12.
    Mariot, L., Picek, S., Leporati, A., Jakobovic, D.: Cellular automata based s-boxes. Cryptogr. Commun. (2018).  https://doi.org/10.1007/s12095-018-0311-8
  13. 13.
    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)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Richardson, D.: Tessellations with local transformations. J. Comput. Syst. Sci. 6(5), 373–388 (1972)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Stinson, D.R.: Combinatorial Designs - Constructions and Analysis. Springer, New York (2004).  https://doi.org/10.1007/b97564CrossRefzbMATHGoogle Scholar
  17. 17.
    Sutner, K.: De Bruijn graphs and linear cellular automata. Complex Syst. 5(1), 19–30 (1991)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Tompa, M., Woll, H.: How to share a secret with cheaters. J. Cryptol. 1(2), 133–138 (1988)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi Milano-BicoccaMilanItaly

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