  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 up to local
rules of diameter $d=5$, and then focusing on the subclass of linear
bipermutive rules. In this case, we characterize 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 of maximal period up to diameter
$d=11$.
