Digital image scrambling based on elementary cellular automata

Three-state von Neumann cellular automata and pattern generation

This paper proposes an unique image encryption scheme in the hybrid domain based on Logistic Map, Reversible Integer Wavelet Transform, and Elementary Cellular Automata (CA). In the proposed work, position permutation is done using Logistic Map equation in the spatial domain and pixel value transformation is done using CA rule and randomly generated matrix created using Logistic map equation in the Transfer Domain with the help of Reversible Integer Wavelet Transform. This paper handouts a pristine application of Logistic map functions and CA in the transfer domain and in spatial domain. Analysis shows that the relationship among the transfer domain coefficients is greatly reduced and the randomness metric is equal to 7.999 much closer to the ideal value of 8.

This paper explores the concept of isomorphism in cellular automata (CAs), focusing on identifying and understanding isomorphic relationships between distinct CAs. A cellular automaton (CA) is said to be isomorphic to another CA if the configuration transition diagrams of the two CAs are identical. To illustrate how isomorphism entails the preservation of structural properties while allowing for variations in state representations, we exhaustively explore the isomorphic reversible rules in the first degree CAs (FDCAs), that is, the simple three-neighborhood CAs represented by eight parameters. The experimental study includes reversible FDCAs for states two to 10 with cell lengths of four to seven under the null boundary condition and states two to seven with cell lengths of four to seven under the periodic boundary condition. Given a CA, we also provide two synthesis methods for identifying the set of CAs that are isomorphic to it. The first method operates by permuting the states of a group of CA cell locations, whereas the second method involves exchanging cells at various points. We demonstrate that, depending on the range of exchanged cell positions, the isomorphic CAs produced in this way may have a bigger neighborhood than the input CA. Finally, we provide an exhaustive classification of the elementary CAs (ECAs) using the first synthesis method and compare it with existing results.

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.

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.

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.

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.

The coarse-graining is an approximation procedure widely used for simplification of mathematical and numerical models of multiscale systems. It reduces superfluous – microscopic – degrees of freedom. Israeli and Goldenfeld demonstrated in [1,2] that the coarse-graining can be employed for elementary cellular automata (CA), producing interesting interdependences between them. However, extending their investigation on more complex CA rules appeared to be impossible due to the high computational complexity of the coarse-graining algorithm. We demonstrate here that this complexity can be substantially decreased. It allows for scrutinizing much broader class of cellular automata in terms of their coarse graining. By using our algorithm we found out that the ratio of the numbers of elementary CAs having coarse grained representation to “degenerate” – irreducible – cellular automata, strongly increases with increasing the “grain” size of the approximation procedure. This rises principal questions about the formal limits in modeling of realistic multiscale systems.Keywordsmultiscale systemscoarse grainingcellular automata

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.

Non-linear oscillators with Kuramoto-like local coupling: Complexity analysis and spatiotemporal pattern generation

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.

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.

This work investigates elementary cellular automata comparing their dynamic evolution on two topologies: the usual regular lattice and a small-world network constructed as a directed graph. The number of iterations for any initial perturbation to propagate over all cells of the automaton corresponds to how quickly an initial configuration reaches a high entropy state, and the temporal rate of propagation of the perturbation can be evaluated from a chaotic set of elementary cellular automata rules using an entropy measure. Results indicate that the average entropy can be nearly tripled in the small-world network topology, suggesting much faster applications (e.g. in Cryptography) from alterations in the topological arrangement of usual cellular automata regular lattices.

Cellular Automata (CA) have a long history being employed as pseudo-random number generators (PRNG), especially for cryptographic applications such as keystream generation in stream ciphers. Initially starting from the study of rule 30 of elementary CA, multiple rules where the objects of investigation and were shown to be able to pass most of the rigorous statistical tests used to assess the quality of PRNG. In all cases, the CA employed where of the classical, synchronous kind. This assumes a global clock regulating all CA updates which can be a weakness if an attacker is able to tamper it. Here we study how much asynchrony is necessary to make a CA-based PRNG ineffective. We have found that elementary CA are subdivided into three class: (1) there is a “state transition” where, after a certain level of asynchrony, the CA loses the ability to generate strong random sequences, (2) the randomness of the sequences increases with a limited level of asynchrony, or (3) CA normally unable to be used as PRNG exhibit a much stronger ability to generate random sequences when asynchrony is introduced.

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.