arXiv subjects

Luca Mariot

Publications and source records attributed to Luca Mariot.

At least 19 recordsLinked to original sources

arXiv2026

On the transversals of Latin squares generated by nonlinear bipermutive cellular automata

In this short paper, we begin to investigate the conditions under which a generic Bipermutive Cellular Automaton (BCA) with no-boundary conditions of diameter $d$ generates a Latin square of order $N=2^{d-1}$ admitting an orthogonal mate, without relying on the linearity of the local rule. Since an orthogonal mate exists if and only if the Latin square can be partitioned into $N$ disjoint \emph{transversals}, we start by characterizing the subclass of BCA whose Latin squares have a transversal on the main diagonal. In particular, we prove that the main diagonal forms a transversal if and only if the generating function of the bipermutive local rule induces an invertible CA with periodic boundary conditions on a configuration of size $d-1$. We then perform exhaustive search experiments, showing that $d=6$ is the smallest diameter for which there exist nonlinear bipermutive CA that generate Latin squares with a transversal on the main diagonal.

cs.FL↗

arXiv2026

Monotone but Exciting: On Evolving Monotone Boolean Functions with High Nonlinearity

Monotone Boolean functions are a structurally important class of Boolean functions, but their restricted form imposes strong limitations on achievable nonlinearity. In this paper, we investigate whether evolutionary computation can evolve monotone Boolean functions with high nonlinearity, both in the balanced and imbalanced settings. We consider three solution encodings: the standard truth table representation, a balanced truth table encoding that preserves Hamming weight, and a symbolic tree-based genetic programming representation. To guide the search toward monotone increasing functions, we introduce a non-monotonicity penalty and combine it with fitness functions targeting balancedness and nonlinearity. Experimental results are reported for dimensions from $n=5$ to $n=14$. The results show that evolutionary search can discover monotone Boolean functions with nonlinearities clearly exceeding those of majority functions, and in several cases approaching the best currently known values for monotone functions. At the same time, the experiments reveal substantial differences between encodings: the balanced truth table encoding performs poorly for larger dimensions, while the standard truth table and genetic programming encodings remain competitive, with genetic programming becoming especially relevant in the largest tested dimensions.

cs.NE↗

arXiv2026

How to reconstruct (anonymously) a secret cellular automaton

We consider threshold secret sharing schemes based on cellular automata (CA) that allows for anonymous reconstruction, meaning that the secret can be recovered only as a function of the shares, without knowing the participants' identities. To this end, we revisit the basic characterization of $(2,n)$ threshold schemes based on CA in terms of Mutually Orthogonal Latin Squares (MOLS), and redefine the secret space as the MOLS family itself, showing that the new resulting scheme enables anonymous reconstruction of secret CA rules. Finally, we discuss the trade-off between the number of secret CA that can be shared and the computational complexity of the recovery phase.

cs.CR↗

arXiv2026

IDEM Enough? Evolving Highly Nonlinear Idempotent Boolean Functions

Idempotent Boolean functions form a highly structured subclass of Boolean functions that is closely related to rotation symmetry under a normal-basis representation and to invariance under a fixed linear map in a polynomial basis. These functions are attractive as candidates for cryptographic design, yet their additional algebraic constraints make the search for high nonlinearity substantially more difficult than in the unconstrained case. In this work, we investigate evolutionary methods for constructing highly nonlinear idempotent Boolean functions for dimensions $n=5$ up to $n=12$ using a polynomial basis representation with canonical primitive polynomials. Our results show that the problem of evolving idempotent functions is difficult due to the disruptive nature of crossover and mutation operators. Next, we show that idempotence can be enforced by encoding the truth table on orbits, yielding a compact genome of size equal to the number of distinct squaring orbits.

cs.CR↗

arXiv2026

NegaBent, No Regrets: Evolving Spectrally Flat Boolean Functions

Negabent Boolean functions are defined by having a flat magnitude spectrum under the nega-Hadamard transform. They exist in both even and odd dimensions, and the subclass of functions that are simultaneously bent and negabent (bent-negabent) has attracted interest due to the combined optimal periodic and negaperiodic spectral properties. In this work, we investigate how evolutionary algorithms can be used to evolve (bent-)negabent Boolean functions. Our experimental results indicate that evolutionary algorithms, especially genetic programming, are a suitable approach for evolving negabent Boolean functions, and we successfully evolve such functions in all dimensions we consider.

cs.NE↗

arXiv2025

On Counts and Densities of Homogeneous Bent Functions: An Evolutionary Approach

Boolean functions with strong cryptographic properties, such as high nonlinearity and algebraic degree, are important for the security of stream and block ciphers. These functions can be designed using algebraic constructions or metaheuristics. This paper examines the use of Evolutionary Algorithms (EAs) to evolve homogeneous bent Boolean functions, that is, functions whose algebraic normal form contains only monomials of the same degree and that are maximally nonlinear. We introduce the notion of density of homogeneous bent functions, facilitating the algorithmic design that results in finding quadratic and cubic bent functions in different numbers of variables.

cs.NE↗

arXiv2025

A Systematic Study on the Design of Odd-Sized Highly Nonlinear Boolean Functions via Evolutionary Algorithms

This paper focuses on the problem of evolving Boolean functions of odd sizes with high nonlinearity, a property of cryptographic relevance. Despite its simple formulation, this problem turns out to be remarkably difficult. We perform a systematic evaluation by considering three solution encodings and four problem instances, analyzing how well different types of evolutionary algorithms behave in finding a maximally nonlinear Boolean function. Our results show that genetic programming generally outperforms other evolutionary algorithms, although it falls short of the best-known results achieved by ad-hoc heuristics. Interestingly, by adding local search and restricting the space to rotation symmetric Boolean functions, we show that a genetic algorithm with the bitstring encoding manages to evolve a $9$-variable Boolean function with nonlinearity 241.

cs.NE↗

arXiv2025

Self-Orthogonal Cellular Automata

It is known that no-boundary Cellular Automata (CA) defined by bipermutive local rules give rise to Latin squares. In this paper, we study under which conditions the Latin square generated by a bipermutive CA is self-orthogonal, i.e. orthogonal to its transpose. We first enumerate all bipermutive CA over the binary alphabet up to diameter $d=6$, remarking that only some linear rules give rise to self-orthogonal Latin squares. We then give a full theoretical characterization of self-orthogonal linear CA, by considering the square matrix obtained by stacking the transition matrices of the CA and of its transpose, and determining when it is invertible. Interestingly, the stacked matrix turns out to have a circulant structure, for which there exists an extensive body of results to characterize its invertibility. Further, for the case of the binary alphabet we prove that irreducibility is a sufficient condition for self-orthogonality, and we derive a simpler characterization which boils down to computing the parity of the central coefficients of the local rule.

cs.DM↗

arXiv2025

Combinatorial Designs and Cellular Automata: A Survey

Cellular Automata (CA) are commonly investigated as a particular type of dynamical systems, defined by shift-invariant local rules. In this paper, we consider instead CA as algebraic systems, focusing on the combinatorial designs induced by their short-term behavior. Specifically, we review the main results published in the literature concerning the construction of mutually orthogonal Latin squares via bipermutive CA, considering both the linear and nonlinear cases. We then survey some significant applications of these results to cryptography, and conclude with a discussion of open problems to be addressed in future research on CA-based combinatorial designs.

math.CO↗

arXiv2025

Degree is Important: On Evolving Homogeneous Boolean Functions

Boolean functions with good cryptographic properties like high nonlinearity and algebraic degree play an important in the security of stream and block ciphers. Such functions may be designed, for instance, by algebraic constructions or metaheuristics. This paper investigates the use of Evolutionary Algorithms (EAs) to design homogeneous bent Boolean functions, i.e., functions that are maximally nonlinear and whose algebraic normal form contains only monomials of the same degree. In our work, we evaluate three genotype encodings and four fitness functions. Our results show that while EAs manage to find quadratic homogeneous bent functions (with the best method being a GA leveraging a restricted encoding), none of the approaches result in cubic homogeneous bent functions.

cs.NE↗

arXiv2024

The More the Merrier: On Evolving Five-valued Spectra Boolean Functions

Evolving Boolean functions with specific properties is an interesting optimization problem since, depending on the combination of properties and Boolean function size, the problem can range from very simple to (almost) impossible to solve. Moreover, some problems are more interesting as there may be only a few options for generating the required Boolean functions. This paper investigates one such problem: evolving five-valued spectra Boolean functions, which are the functions whose Walsh-Hadamard coefficients can only take five distinct values. We experimented with three solution encodings, two fitness functions, and 12 Boolean function sizes and showed that the tree encoding is superior to other choices, as we can obtain five-valued Boolean functions with high nonlinearity.

cs.NE↗

arXiv2024

On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors

We consider the construction of maximal families of polynomials over the finite field $\mathbb{F}_q$, all having the same degree $n$ and a nonzero constant term, where the degree of the GCD of any two polynomials is $d$ with $1 \le d\le n$. The motivation for this problem lies in a recent construction for subspace codes based on cellular automata. More precisely, the minimum distance of such subspace codes relates to the maximum degree $d$ of the pairwise GCD in this family of polynomials. Hence, characterizing the maximal families of such polynomials is equivalent to determining the maximum cardinality of the corresponding subspace codes for a given minimum distance. We first show a lower bound on the cardinality of such families, and then focus on the specific case where $d=1$. There, we characterize the maximal families of polynomials over the binary field $\mathbb{F}_2$. Our findings prompt several more open questions, which we plan to address in an extended version of this work.

cs.DM↗

arXiv2024

Insights Gained after a Decade of Cellular Automata-based Cryptography

Cellular Automata (CA) have been extensively used to implement symmetric cryptographic primitives, such as pseudorandom number generators and S-boxes. However, most of the research in this field, except the very early works, seems to be published in non-cryptographic venues. This phenomenon poses a problem of relevance: are CA of any use to cryptographers nowadays? This paper provides insights into this question by briefly outlining the history of CA-based cryptography. In doing so, the paper identifies some shortcomings in the research addressing the design of symmetric primitives exclusively from a CA standpoint, alongside some recommendations for future research. Notably, the paper remarks that researchers working in CA and cryptography often tackle similar problems, albeit under different perspectives and terminologies. This observation indicates that there is still ample room for fruitful collaborations between the CA and cryptography communities in the future.

cs.CR↗

arXiv2024

A Systematic Evaluation of Evolving Highly Nonlinear Boolean Functions in Odd Sizes

Boolean functions are mathematical objects used in diverse applications. Different applications also have different requirements, making the research on Boolean functions very active. In the last 30 years, evolutionary algorithms have been shown to be a strong option for evolving Boolean functions in different sizes and with different properties. Still, most of those works consider similar settings and provide results that are mostly interesting from the evolutionary algorithm's perspective. This work considers the problem of evolving highly nonlinear Boolean functions in odd sizes. While the problem formulation sounds simple, the problem is remarkably difficult, and the related work is extremely scarce. We consider three solutions encodings and four Boolean function sizes and run a detailed experimental analysis. Our results show that the problem is challenging, and finding optimal solutions is impossible except for the smallest tested size. However, once we added local search to the evolutionary algorithm, we managed to find a Boolean function in nine inputs with nonlinearity 241, which, to our knowledge, had never been accomplished before with evolutionary algorithms.

cs.NE↗

arXiv2024

A Discrete Particle Swarm Optimizer for the Design of Cryptographic Boolean Functions

A Particle Swarm Optimizer for the search of balanced Boolean functions with good cryptographic properties is proposed in this paper. The algorithm is a modified version of the permutation PSO by Hu, Eberhart and Shi which preserves the Hamming weight of the particles positions, coupled with the Hill Climbing method devised by Millan, Clark and Dawson to improve the nonlinearity and deviation from correlation immunity of Boolean functions. The parameters for the PSO velocity equation are tuned by means of two meta-optimization techniques, namely Local Unimodal Sampling (LUS) and Continuous Genetic Algorithms (CGA), finding that CGA produces better results. Using the CGA-evolved parameters, the PSO algorithm is then run on the spaces of Boolean functions from $n=7$ to $n=12$ variables. The results of the experiments are reported, observing that this new PSO algorithm generates Boolean functions featuring similar or better combinations of nonlinearity, correlation immunity and propagation criterion with respect to the ones obtained by other optimization methods.

cs.NE↗

arXiv2023

A New Angle: On Evolving Rotation Symmetric Boolean Functions

Rotation symmetric Boolean functions represent an interesting class of Boolean functions as they are relatively rare compared to general Boolean functions. At the same time, the functions in this class can have excellent properties, making them interesting for various practical applications. The usage of metaheuristics to construct rotation symmetric Boolean functions is a direction that has been explored for almost twenty years. Despite that, there are very few results considering evolutionary computation methods. This paper uses several evolutionary algorithms to evolve rotation symmetric Boolean functions with different properties. Despite using generic metaheuristics, we obtain results that are competitive with prior work relying on customized heuristics. Surprisingly, we find that bitstring and floating point encodings work better than the tree encoding. Moreover, evolving highly nonlinear general Boolean functions is easier than rotation symmetric ones.

cs.NE↗

arXiv2023

Look into the Mirror: Evolving Self-Dual Bent Boolean Functions

Bent Boolean functions are important objects in cryptography and coding theory, and there are several general approaches for constructing such functions. Metaheuristics proved to be a strong choice as they can provide many bent functions, even when the size of the Boolean function is large (e.g., more than 20 inputs). While bent Boolean functions represent only a small part of all Boolean functions, there are several subclasses of bent functions providing specific properties and challenges. One of the most interesting subclasses comprises (anti-)self-dual bent Boolean functions. This paper provides a detailed experimentation with evolutionary algorithms with the goal of evolving (anti-)self-dual bent Boolean functions. We experiment with two encodings and two fitness functions to directly evolve self-dual bent Boolean functions. Our experiments consider Boolean functions with sizes of up to 16 inputs, and we successfully construct self-dual bent functions for each dimension. Moreover, when comparing with the evolution of bent Boolean functions, we notice that the difficulty for evolutionary algorithms is rather similar. Finally, we also tried evolving secondary constructions for self-dual bent functions, but this direction provided no successful results.

cs.NE↗

arXiv2023

Smooth Number Message Authentication Code in the IoT Landscape

This paper presents the Smooth Number Message Authentication Code (SNMAC) for the context of lightweight IoT devices. The proposal is based on the use of smooth numbers in the field of cryptography, and investigates how one can use them to improve the security and performance of various algorithms or security constructs. The literature findings suggest that current IoT solutions are viable and promising, yet they should explore the potential usage of smooth numbers. The methodology involves several processes, including the design, implementation, and results evaluation. After introducing the algorithm, provides a detailed account of the experimental performance analysis of the SNMAC solution, showcasing its efficiency in real-world scenarios. Furthermore, the paper also explores the security aspects of the proposed SNMAC algorithm, offering valuable insights into its robustness and applicability for ensuring secure communication within IoT environments.

cs.CR↗