arXiv subjects

Luca Manzoni

Publications and source records attributed to Luca Manzoni.

At least 19 recordsLinked to original sources

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↗

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↗

arXiv2024

Computing eulerian magnitude homology

In this paper tackle the problem of computing the ranks of certain eulerian magnitude homology groups of a graph G. First, we analyze the computational cost of our problem and prove that it is #W[1]-complete. Then we develop the first diagonal algorithm, a breadth-first-search-based algorithm parameterized by the diameter of the graph to calculate the ranks of the homology groups of interest. To do this, we leverage the close relationship between the combinatorics of the homology boundary map and the substructures appearing in the graph. We then discuss the feasibility of the presented algorithm and consider future perspectives.

cs.CC↗

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

Fixed Points and Attractors of Reactantless and Inhibitorless Reaction Systems

Reaction systems are discrete dynamical systems that model biochemical processes in living cells using finite sets of reactants, inhibitors, and products. We investigate the computational complexity of a comprehensive set of problems related to the existence of fixed points and attractors in two constrained classes of reaction systems, in which either reactants or inhibitors are disallowed. These problems have biological relevance and have been extensively studied in the unconstrained case; however, they remain unexplored in the context of reactantless or inhibitorless systems. Interestingly, we demonstrate that although the absence of reactants or inhibitors simplifies the system's dynamics, it does not always lead to a reduction in the complexity of the considered problems.

cs.CC↗

arXiv2023

Local Search, Semantics, and Genetic Programming: a Global Analysis

Geometric Semantic Geometric Programming (GSGP) is one of the most prominent Genetic Programming (GP) variants, thanks to its solid theoretical background, the excellent performance achieved, and the execution time significantly smaller than standard syntax-based GP. In recent years, a new mutation operator, Geometric Semantic Mutation with Local Search (GSM-LS), has been proposed to include a local search step in the mutation process based on the idea that performing a linear regression during the mutation can allow for a faster convergence to good-quality solutions. While GSM-LS helps the convergence of the evolutionary search, it is prone to overfitting. Thus, it was suggested to use GSM-LS only for a limited number of generations and, subsequently, to switch back to standard geometric semantic mutation. A more recently defined variant of GSGP (called GSGP-reg) also includes a local search step but shares similar strengths and weaknesses with GSM-LS. Here we explore multiple possibilities to limit the overfitting of GSM-LS and GSGP-reg, ranging from adaptive methods to estimate the risk of overfitting at each mutation to a simple regularized regression. The results show that the method used to limit overfitting is not that important: providing that a technique to control overfitting is used, it is possible to consistently outperform standard GSGP on both training and unseen data. The obtained results allow practitioners to better understand the role of local search in GSGP and demonstrate that simple regularization strategies are effective in controlling overfitting.

cs.NE↗

arXiv2023

Frequency maps reveal the correlation between Adversarial Attacks and Implicit Bias

Despite their impressive performance in classification tasks, neural networks are known to be vulnerable to adversarial attacks, subtle perturbations of the input data designed to deceive the model. In this work, we investigate the correlation between these perturbations and the implicit bias of neural networks trained with gradient-based algorithms. To this end, we analyse a representation of the network's implicit bias through the lens of the Fourier transform. Specifically, we identify unique fingerprints of implicit bias and adversarial attacks by calculating the minimal, essential frequencies needed for accurate classification of each image, as well as the frequencies that drive misclassification in its adversarially perturbed counterpart. This approach enables us to uncover and analyse the correlation between these essential frequencies, providing a precise map of how the network's biases align or contrast with the frequency components exploited by adversarial attacks. To this end, among other methods, we use a newly introduced technique capable of detecting nonlinear correlations between high-dimensional datasets. Our results provide empirical evidence that the network bias in Fourier space and the target frequencies of adversarial attacks are highly correlated and suggest new potential strategies for adversarial defence.

cs.LG↗

arXiv2023

A classification of S-boxes generated by Orthogonal Cellular Automata

Most of the approaches published in the literature to construct S-boxes via Cellular Automata (CA) work by either iterating a finite CA for several time steps, or by a one-shot application of the global rule. The main characteristic that brings together these works is that they employ a single CA rule to define the vectorial Boolean function of the S-box. In this work, we explore a different direction for the design of S-boxes that leverages on Orthogonal CA (OCA), i.e. pairs of CA rules giving rise to orthogonal Latin squares. The motivation stands on the facts that an OCA pair already defines a bijective transformation, and moreover the orthogonality property of the resulting Latin squares ensures a minimum amount of diffusion. We exhaustively enumerate all S-boxes generated by OCA pairs of diameter $4 \le d \le 6$, and measure their nonlinearity. Interestingly, we observe that for $d=4$ and $d=5$ all S-boxes are linear, despite the underlying CA local rules being nonlinear. The smallest nonlinear S-boxes emerges for $d=6$, but their nonlinearity is still too low to be used in practice. Nonetheless, we unearth an interesting structure of linear OCA S-boxes, proving that their Linear Components Space (LCS) is itself the image of a linear CA, or equivalently a polynomial code. We finally classify all linear OCA S-boxes in terms of their generator polynomials.

cs.CR↗

arXiv2022

Evolutionary Strategies for the Design of Binary Linear Codes

The design of binary error-correcting codes is a challenging optimization problem with several applications in telecommunications and storage, which has also been addressed with metaheuristic techniques and evolutionary algorithms. Still, all these efforts focused on optimizing the minimum distance of unrestricted binary codes, i.e., with no constraints on their linearity, which is a desirable property for efficient implementations. In this paper, we present an Evolutionary Strategy (ES) algorithm that explores only the subset of linear codes of a fixed length and dimension. To that end, we represent the candidate solutions as binary matrices and devise variation operators that preserve their ranks. Our experiments show that up to length $n=14$, our ES always converges to an optimal solution with a full success rate, and the evolved codes are all inequivalent to the Best-Known Linear Code (BKLC) given by MAGMA. On the other hand, for larger lengths, both the success rate of the ES as well as the diversity of the evolved codes start to drop, with the extreme case of $(16,8,5)$ codes which all turn out to be equivalent to MAGMA's BKLC.

cs.NE↗

arXiv2022

Building Correlation Immune Functions from Sets of Mutually Orthogonal Cellular Automata

Correlation immune Boolean functions play an important role in the implementation of efficient masking countermeasures for side-channel attacks in cryptography. In this paper, we investigate a method to construct correlation immune functions through families of mutually orthogonal cellular automata (MOCA). First, we show that the orthogonal array (OA) associated to a family of MOCA can be expanded to a binary OA of strength at least 2. To prove this result, we exploit the characterization of MOCA in terms of orthogonal labelings on de Bruijn graphs. Then, we use the resulting binary OA to define the support of a second-order correlation immune function. Next, we perform some computational experiments to construct all such functions up to $n=12$ variables, and observe that their correlation immunity order is actually greater, always at least 3. We conclude by discussing how these results open up interesting perspectives for future research, with respect to the search of new correlation-immune functions and binary orthogonal arrays.

cs.CR↗

arXiv2022

The Influence of Local Search over Genetic Algorithms with Balanced Representations

We continue the study of Genetic Algorithms (GA) on combinatorial optimization problems where the candidate solutions need to satisfy a balancedness constraint. It has been observed that the reduction of the search space size granted by ad-hoc crossover and mutation operators does not usually translate to a substantial improvement of the GA performances. There is still no clear explanation of this phenomenon, although it is suspected that a balanced representation might yield a more irregular fitness landscape, where it could be more difficult for GA to converge to a global optimum. In this paper, we investigate this issue by adding a local search step to a GA with balanced operators, and use it to evolve highly nonlinear balanced Boolean functions. In particular, we organize our experiments around two research questions, namely if local search (1) improves the convergence speed of GA, and (2) decreases the population diversity. Surprisingly, while our results answer affirmatively the first question, they also show that adding local search actually \emph{increases} the diversity among the individuals in the population. We link these findings to some recent results on fitness landscape analysis for problems on Boolean functions.

cs.NE↗

arXiv2022

On the Linear Components Space of S-boxes Generated by Orthogonal Cellular Automata

We investigate S-boxes defined by pairs of Orthogonal Cellular Automata (OCA), motivated by the fact that such CA always define bijective vectorial Boolean functions, and could thus be interesting for the design of block ciphers. In particular, we perform an exhaustive search of all nonlinear OCA pairs of diameter $d=4$ and $d=5$, which generate S-boxes of size $6\times 6$ and $8\times 8$, respectively. Surprisingly, all these S-boxes turn out to be linear, and thus they are not useful for the design of confusion layers in block ciphers. However, a closer inspection of these S-boxes reveals a very interesting structure. Indeed, we remark that the linear components space of the OCA-based S-boxes found by our exhaustive search are themselves the kernels of linear CA, or, equivalently, \emph{polynomial codes}. We finally classify the polynomial codes of the S-boxes obtained in our exhaustive search and observe that, in most cases, they actually correspond to the cyclic code with generator polynomial $X^{b}+1$, where $b=d-1$. Although these findings rule out the possibility of using OCA to design good S-boxes in block ciphers, they give nonetheless some interesting insights for a theoretical characterization of nonlinear OCA pairs, which is still an open question in general.

cs.CR↗

arXiv2022

The Effect of Multi-Generational Selection in Geometric Semantic Genetic Programming

Among the evolutionary methods, one that is quite prominent is Genetic Programming, and, in recent years, a variant called Geometric Semantic Genetic Programming (GSGP) has shown to be successfully applicable to many real-world problems. Due to a peculiarity in its implementation, GSGP needs to store all the evolutionary history, i.e., all populations from the first one. We exploit this stored information to define a multi-generational selection scheme that is able to use individuals from older populations. We show that a limited ability to use "old" generations is actually useful for the search process, thus showing a zero-cost way of improving the performances of GSGP.

cs.NE↗

arXiv2021

Heuristic Search of (Semi-)Bent Functions based on Cellular Automata

An interesting thread in the research of Boolean functions for cryptography and coding theory is the study of secondary constructions: given a known function with a good cryptographic profile, the aim is to extend it to a (usually larger) function possessing analogous properties. In this work, we continue the investigation of a secondary construction based on cellular automata, focusing on the classes of bent and semi-bent functions. We prove that our construction preserves the algebraic degree of the local rule, and we narrow our attention to the subclass of quadratic functions, performing several experiments based on exhaustive combinatorial search and heuristic optimization through Evolutionary Strategies (ES). Finally, we classify the obtained results up to permutation equivalence, remarking that the number of equivalence classes that our CA-XOR construction can successfully extend grows very quickly with respect to the CA diameter.

cs.CR↗

arXiv2021

Salp Swarm Optimization: a Critical Review

In the crowded environment of bio-inspired population-based metaheuristics, the Salp Swarm Optimization (SSO) algorithm recently appeared and immediately gained a lot of momentum. Inspired by the peculiar spatial arrangement of salp colonies, which are displaced in long chains following a leader, this algorithm seems to provide an interesting optimization performance. However, the original work was characterized by some conceptual and mathematical flaws, which influenced all ensuing papers on the subject. In this manuscript, we perform a critical review of SSO, highlighting all the issues present in the literature and their negative effects on the optimization process carried out by this algorithm. We also propose a mathematically correct version of SSO, named Amended Salp Swarm Optimizer (ASSO) that fixes all the discussed problems. We benchmarked the performance of ASSO on a set of tailored experiments, showing that it is able to achieve better results than the original SSO. Finally, we performed an extensive study aimed at understanding whether SSO and its variants provide advantages compared to other metaheuristics. The experimental results, where SSO cannot outperform simple well-known metaheuristics, suggest that the scientific community can safely abandon SSO.

cs.NE↗

arXiv2021

Parallel Sandpiles or Spurious Bidirectional Icepiles?

In a recent paper E. Formenti and K. Perrot (FP) introduce a global rule assumed to describe the discrete time dynamics associated with a sandpile model under the parallel application of a suitable local rule acting on d dimensional lattices of cells equipped with uniform neighborhood. In this paper we submit this approach to a critical analysis, in the simplest elementary particular case of a one-dimensional lattice, which can be divided in two parts. In the first part we prove that the FP global rule does not describe the dynamics of standard sandpiles, but rather furnishes a description of the quite different situation of height difference between consecutive piles. This is a semantic uncorrect difference of interpretation. In the second part we investigate the consequences of the uncorrect FP assumption proving that their global rule describes a bidirectional spurious dynamics of icepiles (rather than sandpiles), in the sense that this latter is the consequence of application of three local rules: bidirectional vertical rule, bidirectional horizontal rule (typical of icepiles), and a granule jump from the bottom to the top (spurious rule of the dynamics).

cs.FL↗

arXiv2021

CoInGP: Convolutional Inpainting with Genetic Programming

We investigate the use of Genetic Programming (GP) as a convolutional predictor for missing pixels in images. The training phase is performed by sweeping a sliding window over an image, where the pixels on the border represent the inputs of a GP tree. The output of the tree is taken as the predicted value for the central pixel. We consider two topologies for the sliding window, namely the Moore and the Von Neumann neighborhood. The best GP tree scoring the lowest prediction error over the training set is then used to predict the pixels in the test set. We experimentally assess our approach through two experiments. In the first one, we train a GP tree over a subset of 1000 complete images from the MNIST dataset. The results show that GP can learn the distribution of the pixels with respect to a simple baseline predictor, with no significant differences observed between the two neighborhoods. In the second experiment, we train a GP convolutional predictor on two degraded images, removing around 20% of their pixels. In this case, we observe that the Moore neighborhood works better, although the Von Neumann neighborhood allows for a larger training set.

cs.NE↗

arXiv2020

Exploring Semi-bent Boolean Functions Arising from Cellular Automata

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.

nlin.CG↗