Enumeration of Maximal Cycles Generated by Orthogonal Cellular Automata

TL;DR

This paper investigates the cycle structure of orthogonal cellular automata (OCA) for pseudorandom number generation, providing new enumeration algorithms and characterizations to improve understanding of their dynamical behavior.

Contribution

It introduces an empirical analysis of maximal cycles in OCA pairs, characterizes linear OCA cycle structures, and develops algorithms to enumerate OCA with a single maximal cycle.

Findings

Maximal cycles in OCA pairs up to diameter 8 analyzed

Cycle structure characterized for linear OCA using Sylvester matrices

Algorithms devised to enumerate linear OCA with a single maximal cycle

Abstract

Cellular Automata (CA) are an interesting computational model for designing Pseudorandom Number Generators (PRNG), due to the complex dynamical behavior they can exhibit depending on the underlying local rule. Most of the CA-based PRNGs proposed in the literature, however, suffer from poor diffusion since a change in a single cell can propagate only within its neighborhood during a single time step. This might pose a problem especially when such PRNGs are used for cryptographic purposes. In this paper, we consider an alternative approach to generate pseudorandom sequences through \emph{orthogonal CA} (OCA), which guarantees a better amount of diffusion. After defining the related PRNG, we perform an empirical investigation of the maximal cycles in OCA pairs up to diameter $d=8$. Next, we focus on OCA induced by linear rules, giving a characterization of their cycle structure based on the rational canonical form of the associated Sylvester matrix. Finally, we devise an algorithm to enumerate all linear OCA pairs characterized by a single maximal cycle, and apply it up to diameter $d=16$ and $d=13$ for OCA respectively over the binary and ternary alphabets.