Hip to Be (Latin) Square: Maximal Period Sequences from Orthogonal Cellular Automata

Orthogonal Cellular Automata (OCA) have been recently investigated in the literature as a new approach to construct orthogonal Latin squares for cryptographic applications such as secret sharing schemes. In this paper, we consider OCA for a different cryptographic task, namely the generation of pseudorandom sequences. The idea is to iterate a dynamical system where the output of an OCA pair is fed back as a new set of coordinates on the superposed squares. The main advantage is that OCA ensure a certain amount of diffusion in the generated sequences, a property which is usually missing from traditional CA-based pseudorandom number generators. We study the problem of finding OCA pairs with maximal period by first performing an exhaustive search over local rules of diameter up to $\mathbf{d=5}$, and then focusing on the subclass of linear bipermutive rules. In this case, we characterize an upper bound on the periods of the sequences in terms of the order of the subgroup generated by an invertible Sylvester matrix. We finally devise an algorithm based on Lagrange's theorem to efficiently enumerate all linear OCA pairs that induce Sylvester matrices of maximal order up to diameter $d=11$,