Natural Computing

, Volume 16, Issue 3, pp 367–381 | Cite as

Computing the periods of preimages in surjective cellular automata

  • Luca Mariot
  • Alberto Leporati
  • Alberto Dennunzio
  • Enrico Formenti


A basic property of one-dimensional surjective cellular automata (CA) is that any preimage of a spatially periodic configuration (SPC) is spatially periodic as well. This paper investigates the relationship between the periods of SPC and the periods of their preimages for various classes of CA. When the CA is only surjective and y is a SPC of least period p, the least periods of all preimages of y are multiples of p. By leveraging on the De Bruijn graph representation of CA, we devise a general algorithm to compute the least periods appearing in the preimages of a SPC, along with their corresponding multiplicities (i.e. how many preimages have a particular least period). Next, we consider the case of linear and bipermutive cellular automata (LBCA) defined over a finite field as state alphabet. In particular, we show an equivalence between preimages of LBCA and concatenated linear recurring sequences (LRS) that allows us to give a complete characterization of their periods. Finally, we generalize these results to LBCA defined over a finite ring as alphabet.


Cellular automata Surjectivity De Bruijn graph Bipermutivity Linear recurring sequences Linear feedback shift registers 

Mathematics Subject Classification

37B15 68Q80 94A55 



The authors wish to thank Ilkka Törma for suggesting that Lemma 4 holds in the general surjective case, and Marco Previtali for insightful comments about the computational complexity of the u-closure graph building procedure. Further, the authors are grateful to the anonymous reviewers for their helpful comments on how to improve the paper.


  1. Berlekamp ER (1967) Factoring polynomials over finite fields. Bell Syst Tech J 46(8):1853–1859MathSciNetCrossRefzbMATHGoogle Scholar
  2. Cattaneo G, Finelli M, Margara L (2000) Investigating topological Chaos by elementary cellular automata dynamics. Theor Comput Sci 244(1–2):219–241MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cattaneo G, Dennunzio A, Margara L (2004) Solution of some conjectures about topological properties of linear cellular automata. Theor Comput Sci 325(2):249–271MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chassé G (1990) Some remarks on a LFSR ”disturbed” by other sequences. In: EUROCODE ’90, international symposium on coding theory and applications, Udine, Nov 5–9, 1990, Proceedings, pp 215–221Google Scholar
  5. Dennunzio A, Formenti E, Weiss M (2014) Multidimensional cellular automata: closing property, quasi-expansivity, and (un) decidability issues. Theor Comput Sci 516:40–59MathSciNetCrossRefzbMATHGoogle Scholar
  6. Durand B (1999) Global properties of cellular automata. In: Cellular automata and complex systems, Springer, pp 1–22Google Scholar
  7. Formenti E, Papazian C, Scribot PA (2014) Additive flowers. In: CIBB 2014Google Scholar
  8. Hedlund GA (1969) Endomorphisms and automorphisms of the shift dynamical systems. Math Syst Theory 3(4):320–375MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hell M, Johansson T, Maximov A, Meier W (2008) The grain family of stream ciphers. In: New stream cipher designs—the eSTREAM finalists, pp 179–190Google Scholar
  10. Lidl R, Niederreiter H (1994) Introduction to finite fields and their applications. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  11. Mariot L, Leporati A (2014) Sharing secrets by computing preimages of bipermutive cellular automata. In: Cellular automata—11th international conference on cellular automata for Research and industry, ACRI 2014, Krakow, Sept 22–25, 2014. Proceedings, pp 417–426Google Scholar
  12. Mariot L, Leporati A (2015) On the periods of spatially periodic preimages in linear bipermutive cellular automata. In: Cellular automata and discrete complex systems—21st IFIP WG 1.5 international workshop, AUTOMATA 2015, Turku, June 8–10, 2015. Proceedings, pp 181–195Google Scholar
  13. Massey JL (1969) Shift-register synthesis and BCH decoding. IEEE Trans Inf Theory 15(1):122–127MathSciNetCrossRefzbMATHGoogle Scholar
  14. McEliece R (2002) The theory of information and coding. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  15. Perrin D, Pin JÉ (2004) Infinite words: automata, semigroups, logic and games, vol 141. Academic, CambridgezbMATHGoogle Scholar
  16. Sutner K (1991) De Bruijn graphs and linear cellular automata. Complex Syst 5(1):19–30MathSciNetzbMATHGoogle Scholar
  17. Sutner K (2010) Cellular automata, decidability and phasespace. Fundam Inform 104(1–2):141–160MathSciNetzbMATHGoogle Scholar
  18. Wagstaff S (2002) Cunningham project. Accessed 22 July 2016

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

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

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