Non-uniform nonlinear cellular automata with large cycles and their application in pseudo-random number generation

This paper reports a stochastic method of synthesizing three-neighborhood two-state nonlinear non-uniform reversible cellular automata (CAs) having large cycle(s). To develop this method, we propose a parameter which measures the dependency of each cell on its neighbors. Using this parameter, we classify the possible rules of reversible CAs into four classes — completely dependent, partially dependent, weakly dependent and independent. A CA is expected to have larger cycle length if most of the rules of the CA are sensitive to their neighbors; hence, possess higher parameter value. It is shown that our synthesis algorithm, where rules are selected from the first three classes based on Gaussian distribution, can successfully generate CAs having desired length cycle. We empirically show that most of the synthesized CAs have a cycle at least as large as 2n−1, where n is the lattice size. Then, as an application of such CAs, a scheme is developed to design window-based pseudo-random number generators (PRNGs) using (some of) them. These PRNGs are easily implementable on hardware and are highly portable. Finally, when compared on the same platform, our PRNGs outperform most of the well-known PRNGs existing today.