Computing the periods of preimages in surjective cellular automata
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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.
KeywordsCellular automata Surjectivity De Bruijn graph Bipermutivity Linear recurring sequences Linear feedback shift registers
Mathematics Subject Classification37B15 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.
- 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
- Durand B (1999) Global properties of cellular automata. In: Cellular automata and complex systems, Springer, pp 1–22Google Scholar
- Formenti E, Papazian C, Scribot PA (2014) Additive flowers. In: CIBB 2014Google Scholar
- 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
- 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
- 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
- Wagstaff S (2002) Cunningham project. http://homes.cerias.purdue.edu/~ssw/cun/index.html. Accessed 22 July 2016