Hip to Be (Latin) Square: Maximal Period Sequences from Orthogonal Cellular Automata
Abstract
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 CAbased 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 $d=11$.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.07750
 Bibcode:
 2021arXiv210607750M
 Keywords:

 Computer Science  Cryptography and Security;
 Computer Science  Discrete Mathematics;
 Computer Science  Formal Languages and Automata Theory
 EPrint:
 16 pages, 3 figures, 2 tables