Resilient Vectorial Functions and Cyclic Codes Arising from Cellular Automata

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 errorcorrection 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.