We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order q d−1 , we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field F q are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree n over F q . Finally, we present a construction of MOLS generated by LBCA with irreducible polynomials and prove the maximality of the resulting sets, as well as a lower bound which is asymptotically close to their actual number.

We consider the problem of evolving a particular kind of shift-invariant transformation-namely, Reversible Cellular Automata (RCA) defined by conserved landscape rules-using GA and GP. To this end, we employ three different optimization strategies: a single-objective approach carried out with GA and GP where only the reversibility constraint of marker CA is considered, a multi-objective approach based on GP where both reversibility and the Hamming weight are taken into account, and a lexicographic approach where GP first optimizes only the reversibility property until a conserved landscape rule is obtained, and then maximizes the Hamming weight while retaining reversibility. The results are discussed in the context of three different research questions stemming from exhaustive search experiments on conserved landscape CA, which concern 1) the difficulty of the associated optimization problem for GA and GP, 2) the utility of conserved landscape CA in the domain of cryptography and reversible computing, and 3) the relationship between the reversibility property and the Hamming weight.

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In several combinatorial optimization problems arising in cryptography and design theory, the admissible solutions must often satisfy a balancedness constraint, such as being represented by bitstrings with a fixed number of ones. For this reason, several works in the literature tackling these optimization problems with Genetic Algorithms (GA) introduced new balanced crossover operators which ensure that the offspring has the same balancedness characteristics of the parents. However, the use of such operators has never been thoroughly motivated, except for some generic considerations about search space reduction.In this paper, we undertake a rigorous statistical investigation on the effect of balanced and unbalanced crossover operators against three optimization problems from the area of cryptography and coding theory: nonlinear balanced Boolean functions, binary Orthogonal Arrays (OA) and bent functions. In particular, we consider three different balanced crossover operators (each with two variants: "left-to-right" and "shuffled"), two of which have never been published before, and compare their performances with classic one-point crossover. We are able to confirm that the balanced crossover operators performs better than all three balanced crossover operators. Furthermore, in two out of three crossovers, the "left-to-right" version performs better than the "shuffled" version.

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Part 2: Regular PapersInternational audienceWe 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