A cellular automaton-based technique for modeling mesoscale damage evolution

Three-state von Neumann cellular automata and pattern generation

In this paper, we propose a new method to obtain the transition rules of two-dimensional cellular automata (CA) that performs grayscale image processing. CA has the advantages of producing complex systems from the local interaction of simple elements, and has attracted increased research interest. The difficulty of designing CA's transition rules to perform a particular task has severely limited their applications. So, the evolutionary design of CA rules has been studied. In this method, an evolutionary algorithm was used to evolve CA. In recent years, this method has been applied to image processing. Rosin has studied the evolutionary design of two-dimensional CA to perform noise reduction, thinning and convex hulls. Batouche et al. and Slatnia et al. employed genetic algorithm to investigate the possibility of CA to perform edge detection. In the previous methods, 2-state CA was used for binary image processing. Unlike the previous methods, the present method uses 256-state CA rules to perform grayscale image processing. Gene Expression Programming (GEP) proposed by Ferreira is employed as a learning algorithm in which the chromosomes encode the transition rules as expression trees. Experimental results for the reduction of impulse noise, salt-and-pepper noise and gaussian noise show that the proposed method is equivalent to previous methods in performance and more than 100 times faster than the method proposed by Rosin. We show that the rule obtained by the proposed method employs symmetry-based strategy in the noise reduction process and this property can reduce complexity of CA.

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≤d≤6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4 \\le d \\le 6$$\\end{document}, and measure their nonlinearity. Interestingly, we observe that for d=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=4$$\\end{document} and d=5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=5$$\\end{document} all S-boxes are linear, despite the underlying CA local rules being nonlinear. The smallest nonlinear S-boxes emerges for d=6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=6$$\\end{document}, 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 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.

Numerous studies can be found in literature concerning the idea of learning cellular automata (CA) rules that perform a given task by means of machine learning methods. Among these methods, genetic algorithms (GAs) have often been used with excellent results. Nevertheless, few attention has been dedicated so far to the generality and robustness of the learned rules. In this paper, we show that when GAs are used to evolve asynchronous one-dimensional CA rules, they are able to find more general and robust solutions compared to the more usual case of evolving synchronous CA rules.

Cellular automata (CA) with given evolution rules have been widely investigated, but the inverse problem of extracting CA rules from observed data is less studied. Current CA rule extraction approaches are both time consuming and inefficient when selecting neighborhoods. We give a novel approach to identifying CA rules from observed data and selecting CA neighborhoods based on the identified CA model. Our identification algorithm uses a model linear in its parameters and gives a unified framework for representing the identification problem for both deterministic and probabilistic CA. Parameters are estimated based on a minimum variance criterion. An incremental procedure is applied during CA identification to select an initial coarse neighborhood. Redundant cells in the neighborhood are then removed based on parameter estimates, and the neighborhood size is determined using the Bayesian information criterion. Experimental results show the effectiveness of our algorithm and that it outperforms other leading CA identification algorithms.

The evolution of fluvial-deltaic system has been one of the important topics in global change research. With the development of spatial science and technology, Remote Sensing(RS), Geographic Information System(GIS) and Global Positioning System(GPS) have been extensively applied to observing, interpreting and analyzing complex processes in the fluvial-deltaic system. However, modeling and predicting has proved very difficult because such system shows a wide range of different types of non-linear responses. Cellular automata (CA) are discrete, dynamical systems that are divided up into small cells with each cell taking a certain state. The basic idea of cellular automata is to reduce a complex system using certain rules into something simpler. In this paper, we present a cellular automata model for coast erosion simulation at large spatial scale. Of the factors which control the coast erosion, wind is most important because it influences both the power and direction of marine dynamics. In the CA model presented here, coast cells can be eroded by their neighboring marine cells. Due to wind effects, the erosion rate at some direction is larger than that at other direction. Such anisotropy erosion was represented by adding different probability weighting for each direction in CA rules. The model was used to simulate the erosion process of the Qingshuigou spit in the Yellow River Delta. The simulation results resemble the actual coastlines extracted from multitemporal TM images to great degree.

There Can Be Only One: Reversible Cellular Automata and the Conservation of Genki

This research presents a unified, quantitative framework for analyzing the complexity and computational potential of Cellular Automata (CA). By combining the Complexity Score (CS) with the Behavioral Index (BI), the method systematically identifies rules that generate both complex structures and non-trivial interactions - features that recur across known constructions of Turing-complete systems and motivate their use as practical search criteria. This methodology provides a reproducible approach to estimating unbiased simulation parameters, ensuring robust measurements regardless of grid size, time step, or random seed. This reduces variability and enhances comparability across different CA rules and dimensions. The framework applies to all N-dimensional CAs, highlighting edge-of-chaos characteristics. Behavioral analysis allows detailed tracking of object dynamics and collision interactions, providing insights that are not captured by structural metrics alone. Overall, this approach lays a foundation for scalable, quantitative exploration of CA rule spaces and for the systematic identification of candidate universal automata, enabling efficient studies in this area and advancing practical uses of CAs.

Cellular Automata (CA) is used in pseudo-random number generators (PRNG) to produce high-rate random numbers. The randomness of such numbers directly depends upon the CA rules and the number of neighbor cells. The two-dimensional (2D) CA has several limitations such as, finding the best CA rules, boundary cell problems, etc. To address the problems existing in the 2D CA, we propose a random number generator based on 3D cellular automata. The proposed generator is based on rule numbers 43, 85, 170, and 201, and incremental boundary conditions. The rules are randomly changed after each cycle depending on a clock bit mechanism. The output bits are then passed to the Diehard and ENT test suites to test its randomness [1]. The results reveal that the bit stream generated by the proposed scheme passes all the tests present in the test suites.

We illustrate growth phenomena in two-dimensional cellular automata (CA) by four case studies. The first CA, which we call Obstacle Course, describes the effect that obstacles have on such features of simple growth models as linear expansion and coherent asymptotic shape. Our next CA is random-walk-based Internal Diffusion Limited Aggregation, which spreads sublinearly, but with a shape which can be explicitly computed due to hydrodynamic effects. Then we propose a simple scheme for characterizing CA according to their growth properties, as indicated by two Larger than Life examples. Finally, a very simple case of Spatial Prisoner’s Dilemma illustrates nucleation analysis of CA. In essence, analysis of growth models is an attempt to study properties of physical systems far from equilibrium (e.g., Meakin [34] and more than 1300 references cited in the latter). Cellular automata (CA) growth models, by virtue of their simplicity and amenability to computer experimentation [25], have become particularly popular in the last 20 years, especially in physics research literature [40, 42]. Needless to say, precise mathematical results are hard to come by, and many basic questions remain completely open at the rigorous level. The purpose of this chapter, then, is to outline some successes of the mathematical approach and to identify some fundamental difficulties. We will mainly address three themes which can be summarized by the terms: aggregation, nucleation, and constraint-expansion transition. These themes also provide opportunities to touch on the roles of randomness, monotonicity, and linearity in CA investigations. We choose to illustrate these issues by particular CA rules, with little attempt to formulate a general theory. Simplicity is often, and rightly, touted as an important selling point of cellular automata. We have, therefore, tried to choose the simplest models which, while being amenable to some mathematical analysis, raise a host of intriguing unanswered questions. The next few paragraphs outline subsequent sections of this chapter. Aggregation models typically study properties of growth from a small initial seed. Arguably, the simplest dynamics are obtained by adding sites on the boundary in a uniform fashion.

The paper presents new results concerning application of cellular automata (CA) to secret key cryptography extending results presented in Tomassini and Perrenoud (2000), and Tomassini and Sipper (2000). One dimensional, nonuniform CA is considered as a generator of pseudorandom number sequences (PNS) used in cryptography with the secret key. The quality of PNS highly depends on a set of applied CA rules. To find such rules nonuniform CA with two types of rules is considered. The search of rules is performed with use of an evolutionary technique called cellular programming. As the result of the collective behavior of the discovered set of CA rules very high quality PNS are generated. The quality of PNS outperforms the quality of known one dimensional CA-based PNS generators used in secret key cryptography. The extended set of CA rules which was found makes the cryptography system much more resistant on breaking a cryptography key.

This work focuses on side-channel resilient design strategies for symmetrickey cryptographic primitives targeting lightweight applications. In light of NIST’s lightweight cryptography project, design choices for block ciphers must consider not only security against traditional cryptanalysis, but also side-channel security, while adhering to low area and power requirements. In this paper, we explore design strategies for substitution-permutation network (SPN)-based block ciphers that make them amenable to low-cost threshold implementations (TI) - a provably secure strategy against side-channel attacks. The core building blocks for our strategy are cryptographically optimal 4×4 S-Boxes, implemented via repeated iterations of simple cellular automata (CA) rules. We present highly optimized TI circuits for such S-Boxes, that consume nearly 40% less area and power as compared to popular lightweight S-Boxes such as PRESENT and GIFT. We validate our claims via implementation results on ASIC using 180nm technology. We also present a comparison of TI circuits for two popular lightweight linear diffusion layer choices - bit permutations and MixColumns using almost-maximum-distance-separable (almost-MDS) matrices. We finally illustrate design paradigms that combine the aforementioned TI circuits for S-Boxes and diffusion layers to obtain fully side-channel secure SPN block cipher implementations with low area and power requirements.

One can study geometrical models of image patterns generated by CA evolution. For this, the present paper has investigated theoretical and imaginary investigation of two-dimensional (2D) linear and uniform cellular automata (CA). We study 2D linear CA under special boundary (reflexive and adiabatic) conditions over the binary field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{2}$</tex> , i.e. two states case. We investigate the evolution of image patterns corresponding to the uniform linear rules of 2D CA with the reflexive and adiabatic boundary conditions over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{2}$</tex> . The linear rules of CA can be found to be some image copies of a given first image depending on the special boundary types.

Characterization of 1-d Periodic Boundary Reversible CA

A novel image encryption algorithm based on a hybrid model of deoxyribonucleic acid (DNA), cellular automata (CA) and chaotic system is proposed. CA due to its complex behavior has several applications such as generating random numbers and cryptography. DNA rules, DNA sequence XOR operator and CA rules are used simultaneously to encrypt the plain-image pixels. To determine rule number in DNA sequence and also CA, a 2-dimension logistic map is employed. Experimental results show that the proposed image encryption scheme has a lot of characteristics, including large key space, low correlation of adjacent cipher pixels, and high security, which can effectively protect the security of the encrypted image.