Enumerating Orthogonal Latin Squares Generated by Bipermutive Cellular Automata

Description

We consider the problem of enumerating pairs of bipermutive cellular automata (CA) which generate orthogonal Latin squares. Since the problem has already been settled for bipermutive CA with linear local rules, we address the general case of nonlinear rules, which could be interesting for cryptographic applications such as the design of cheater-immune secret sharing schemes. We first prove that two bipermutive CA generating orthogonal Latin squares must have pairwise balanced local rules. Then, we count the number of pairwise balanced bipermutive Boolean functions and enumerate those which generate orthogonal Latin squares up to $n=6$ variables, classifying them with respect to their nonlinearity values.

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