Chaos and ergodicity are decidable for linear cellular automata over [formula omitted

Abstract

We prove that important properties describing complex behaviours as ergodicity, chaos, topological transitivity, and topological mixing, are decidable for one-dimensional linear cellular automata (LCA) over (Z/mZ)n (Theorem 6 and Corollary 7), a large and important class of cellular automata (CA) which are able to exhibit the complex behaviours of general CA and are used in applications. In particular, we provide a decidable characterization of ergodicity, which is known to be equivalent to all the above mentioned properties, in terms of the characteristic polynomial of the matrix associated with LCA. We stress that the setting of LCA over (Z/mZ)n with n>1 is more expressive, gives rise to much more complex dynamics, and is more difficult to deal with than the already investigated case n=1. The proof techniques from [23,25] used when n=1 for obtaining decidable characterizations of dynamical and ergodic properties can no longer be exploited when n>1 for achieving the same goal. Indeed, in order to get the decision algorithm (Algorithm 1) we need to prove a non trivial result of abstract algebra (Theorem 5) which is also of interest in its own.We also illustrate the impact of our results in real-world applications concerning the important and growing domain of cryptosystems which are often based on one-dimensional LCA over (Z/mZ)n with n>1. As a matter of facts, since cryptosystems have to satisfy the so-called confusion and diffusion properties (ensured by ergodicity and chaos, respectively, of the involved LCA) Algorithm *1 turns out to be an important tool for building chaotic/ergodic one-dimensional linear CA over (Z/mZ)n and, hence, for improving the existing methods based on them.