Reversible Cellular Automata (RCA) are a particular kind of shift-invariant transformations characterized by dynamics composed only of disjoint cycles. They have many applications in the simulation of physical systems, cryptography, and reversible computing. In this work, we formulate the search of a specific class of RCA – namely, those whose local update rules are defined by conserved landscapes – as an optimization problem to be tackled with Genetic Algorithms (GA) and Genetic Programming (GP). In particular, our experimental investigation revolves around three different research questions, which we address through a single-objective, a multi-objective, and a lexicographic approach. In the single-objective approach, we observe that GP can already find an optimal solution in the initial population. This indicates that evolutionary algorithms are not needed when evolving only the reversibility of such CA, and a more efficient method is to generate at random syntactic trees that define the local update rule. On the other hand, GA and GP proved to be quite effective in the multi-objective and lexicographic approach to (1) discover a trade-off between the reversibility and the Hamming weight of conserved landscape rules, and (2) observe that conserved landscape CA cannot be used in symmetric cryptography because their Hamming weight (and thus their nonlinearity) is too low.

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Semi-bent Boolean functions are interesting from a cryptographic standpoint, since they possess several desirable properties such as having a low and flat Walsh spectrum, which is useful to resist linear cryptanalysis. In this paper, we consider the search of semi-bent functions through a construction based on cellular automata (CA). In particular, the construction defines a Boolean function by computing the XOR of all output cells in the CA. Since the resulting Boolean functions have the same algebraic degree of the CA local rule, we devise a combinatorial algorithm to enumerate all quadratic Boolean functions. We then apply this algorithm to exhaustively explore the space of quadratic rules of up to 6 variables, selecting only those for which our CA-based construction always yields semi-bent functions of up to 20 variables. Finally, we filter the obtained rules with respect to their balancedness, and remark that the semi-bent functions generated through our construction by the remaining rules have a constant number of linear structures.

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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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Bent Boolean functions play an important role in the design of secure symmetric ciphers, since they achieve the maximum distance from affine functions allowed by Parseval’s relation. Hyper-bent functions, in turn, are those bent functions which additionally reach maximum distance from all bijective monomial functions, and provide further security towards approximation attacks. Being characterized by a stricter definition, hyper-bent functions are rarer than bent functions, and much more difficult to construct. In this paper, we employ several evolutionary algorithms in order to evolve hyper-bent Boolean functions of various sizes. Our results show that hyper-bent functions are extremely difficult to evolve, since we manage to find such functions only for the smallest investigated size. Interestingly, we are able to identify this difficulty as not lying in the evolution of hyper-bent functions itself, but rather in evolving some of their components, i.e. bent functions. Finally, we present an additional parameter to evaluate the performance of evolutionary algorithms when evolving Boolean functions: the diversity of the obtained solutions.

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Cellular Automata (CA) represent an interesting approach to design Substitution
Boxes (S-boxes) having good cryptographic properties and low implementation costs. From the cryptographic perspective, up to now there have been only ad-hoc studies about specific kinds of CA, the best known example being the χ nonlinear transformation used in Keccak. In this paper, we undertake a systematic investigation of the cryptographic properties of S-boxes defined by CA, proving some upper bounds on their nonlinearity and differential
uniformity. Next, we extend some previous published results about the construction of CAbased S-boxes by means of a heuristic technique, namely Genetic Programming (GP). In particular, we propose a “reverse engineering” method based on De Bruijn graphs to determine whether a specific S-box is expressible through a single CA rule. Then, we use GP to assess if some CA-based S-box with optimal cryptographic properties can be described
by a smaller CA. The results show that GP is able to find much smaller CA rules defining the same reference S-boxes up to the size 7 × 7, suggesting that our method could be used to find more efficient representations of CA-based S-boxes for hardware implementations. Finally, we classify up to affine equivalence all 3 × 3 and 4 × 4 CA-based S-boxes.@en

Boolean functions have a prominent role in many real-world applications, which makes them a very active research domain. Throughout the years, various heuristic techniques proved to be an attractive choice for the construction of Boolean functions with different properties. One of the most important properties is nonlinearity, and in particular maximally nonlinear Boolean functions are also called bent functions. In this paper, instead of considering Boolean functions, we experiment with quaternary functions. The corresponding problem is much more difficult and presents an interesting benchmark as well as realworld applications. The results we obtain show that evolutionary metaheuristics, especially genetic programming, succeed in finding quaternary functions with the desired properties. The obtained results in the quaternary domain can also be translated into the binary domain, in which case this approach compares favorably with the state-of-the-art in Boolean optimization. Our techniques are able to find quaternary bent functions for up to 8 inputs, which corresponds to obtaining Boolean bent functions of 16 inputs.

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Orthogonal Arrays (OA) represent an interesting breed of combinatorial designs that finds applications in several domains such as statistics, coding theory, and cryptography. In this work, we address the problem of constructing binary OA through evolutionary algorithms, an approach which received little attention in the combinatorial designs literature. We focus on the representation of a feasible solution, which we encode as a set of Boolean functions whose truth tables are used as the columns of a binary matrix, and on the design of an appropriate fitness function and variation operators for this problem. We finally present experimental results obtained with genetic algorithms (GA) and genetic programming (GP) on optimizing such fitness function, and compare the performances of these two metaheuristics with respect to the size of the considered problem instances. The experimental results show that GP outperforms GA at handling this type of problem, as it converges to an optimal solution in all considered problem instances but one.

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Disjunct Matrices (DM) are a particular kind of binary matrices which have been especially applied to solve the Non-Adaptive Group Testing (NAGT) problem, where the task is to detect any configuration of t defectives out of a population of N items. Traditionally, the methods used to construct DM leverage on error-correcting codes and other related algebraic techniques. Here, we investigate the use of Evolutionary Algorithms to design DM and two of their generalizations, namely Resolvable Matrices (RM) and Almost Disjunct Matrices (ADM). After discussing the basic encoding used to represent the candidate solutions of our optimization problems, we define three fitness functions, each measuring the deviation of a generic binary matrix from being respectively a DM, an RM or an ADM. Next, we employ Estimation of Distribution Algorithms (EDA), Genetic Algorithms (GA), and Genetic Programming (GP) to optimize these fitness functions. The results show that GP achieves the best performances among the three heuristics, converging to an optimal solution on a wider range of problem instances. Although these results do not match those obtained by other state-of-the-art methods in the literature, we argue that our heuristic approach can generate solutions that are not expressible by currently known algebraic techniques, and sketch some possible ideas to further improve its performance.@en

We investigate the design of Orthogonal Latin Squares (OLS) by means of Genetic Algorithms (GA) and Genetic Programming (GP). Since we focus on Latin squares generated by Cellular Automata (CA), the problem can be reduced to the search of pairs of Boolean functions that give rise to OLS when used as CA local rules. As it is already known how to design CA-based OLS with linear Boolean functions, we adopt the evolutionary approach to address the nonlinear case, experimenting with different encodings for the candidate solutions. In particular, for GA we consider single bitstring, double bitstring and quaternary string encodings, while for GP we adopt a double tree representation. We test the two metaheuristics on the spaces of local rules pairs with n = 7 and n = 8 variables, using two fitness functions. The results show that GP is always able to generate OLS, even if the optimal solutions found with the first fitness function are mostly linear. On the other hand, GA achieves a remarkably lower success rate than GP in evolving OLS, but the corresponding Boolean functions are always nonlinear.

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