Resilient Vectorial Functions and Cyclic Codes Arising from Cellular Automata

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9863)

Abstract

Most of the works concerning cryptographic applications of cellular automata (CA) focus on the analysis of the underlying local rules, interpreted as boolean functions. In this paper, we investigate the cryptographic criteria of CA global rules by considering them as vectorial boolean functions. In particular, we prove that the 1-resiliency property of CA with bipermutive local rules is preserved on the corresponding global rules. We then unfold an interesting connection between linear codes and cellular automata, observing that the generator and parity check matrices of cyclic codes correspond to the transition matrices of linear CA. Consequently, syndrome computation in cyclic codes can be performed in parallel by evolving a suitable linear CA, and the error-correction capability is determined by the resiliency of the global rule. As an example, we finally show how to implement the (7, 4, 3) cyclic Hamming code using a CA of radius \(r=2\).

Keywords

Cellular automata Boolean functions S-boxes Resiliency Linear feedback shift registers Cyclic codes Hamming codes 

References

  1. 1.
    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. Cambridge University Press, New York (2010)Google Scholar
  2. 2.
    Formenti, E., Imai, K., Martin, B., Yunés, J.-B.: Advances on random sequence generation by uniform cellular automata. In: Calude, C.S., Freivalds, R., Kazuo, I. (eds.) Gruska Festschrift. LNCS, vol. 8808, pp. 56–70. Springer, Heidelberg (2014)Google Scholar
  3. 3.
    Koc, C.K., Apohan, A.M.: Inversion of cellular automata iterations. IEE Proc. Comput. Digit. Tech. 144(5), 279–284 (1997). IETCrossRefGoogle Scholar
  4. 4.
    Leporati, A., Mariot, L.: Cryptographic properties of bipermutive cellular automata rules. J. Cell. Aut. 9(5–6), 437–475 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Mariot, L., Leporati, A.: On the periods of spatially periodic preimages in linear bipermutive cellular automata. In: Kari, J. (ed.) AUTOMATA 2015. LNCS, vol. 9099, pp. 181–195. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  6. 6.
    McEliece, R.J.: The Theory of Information and Coding. Cambridge University Press, New York (1985)Google Scholar
  7. 7.
    Meier, W., Staffelbach, O.: Analysis of pseudo random sequences generated by cellular automata. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 186–199. Springer, Heidelberg (1991)Google Scholar
  8. 8.
    Stinson, D.R.: Combinatorial Designs: Constructions and Analysis. Springer, Heidelberg (2004)MATHGoogle Scholar
  9. 9.
    Siegenthaler, T.: Decrypting a class of stream ciphers using ciphertext only. IEEE Trans. Comput. C–34(1), 81–85 (1985)CrossRefGoogle Scholar
  10. 10.
    Wolfram, S.: Cryptography with cellular automata. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 429–432. Springer, Heidelberg (1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi Milano-BicoccaMilanoItaly
  2. 2.Laboratoire I3SUniversité Nice-Sophia AntipolisSophia AntipolisFrance

Personalised recommendations