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

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.@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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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

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.@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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