Cellular automata based S-boxes

The aim of this work is to find large S-Boxes, typically operating on $$8$$8i?źbits, having both good cryptographic properties and a low implementation cost. Such S-Boxes are suitable building-blocks in many lightweight block ciphers since they may achieve a better security level than designs based directly on smaller S-Boxes. We focus on S-Boxes corresponding to three rounds of a balanced Feistel and of a balanced MISTY structure, and generalize the recent results by Li and Wang on the best differential uniformity and linearity offered by such a construction. Most notably, we prove that Feistel networks supersede MISTY networks for the construction of $$8$$8-bit permutations. Based on these results, we also provide a particular instantiation of an $$8$$8-bit permutation with better properties than the S-Boxes used in several ciphers, including Robin, Fantomas or CRYPTON.

Recently, a new structure called butterfly introduced by Perrin et at. is attractive for that it has very good cryptographic properties: the differential uniformity is at most equal to 4 and algebraic degree is also very high when exponent $e=3$. It is conjecture that the nonlinearity is also optimal for every odd $k$, which was proposed as a open problem. In this paper, we further study the butterfly structures and show that these structure with exponent $e=2^i+1$ have also very good cryptographic properties. More importantly, we prove in theory the nonlinearity is optimal for every odd $k$, which completely solve the open problem. Finally, we study the butter structures with trivial coefficient and show these butterflies have also optimal nonlinearity. Furthermore, we show that the closed butterflies with trivial coefficient are bijective as well, which also can be used to serve as a cryptographic primitive.

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.

Three-neighbourhood Cellular Automata (CA) are widely studied and accepted as suitable cryptographic primitive. Rule 30, a 3-neighbourhood nonlinear CA rule, was proposed as an ideal candidate for cryptographic primitive by Wolfram. However, rule 30 was shown to be weak against Meier-Staffelbach attack [11]. The cryptographic properties like diffusion and randomness increase with increase in neighbourhood radius and thus opens the avenue of exploring the cryptographic properties of 4-neighbourhood CA. This work explores whether 4-neighbourhood CA can be a better cryptographic primitive. We construct a class of cryptographically suitable 4-neighbourhood nonlinear CA rules that resembles rule 30 and study its cryptographic properties. Four-neighbourhood nonlinear CA are shown to be resistant against Meier-Staffelbach attack on rule 30, justifying the applicability of 4-neighbourhood CA as better cryptographic primitives.

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.

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.

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.

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.

Modern block ciphers can be used for designing security schemes to meet the security requirements in underwater acoustic sensor networks. The substitution boxes (S-boxes) used in block ciphers must have good cryptographic properties to resist various attacks. The difficulty in the design of S-boxes is that the cryptographic properties of S-boxes have no obvious distribution. In this paper, we investigate the certain subgraph of the Cayley graph of the symmetric group to get some information about the distribution of the cryptographic properties of S-boxes. Based on the information obtained, an algorithm for designing S-boxes with good cryptographic properties is proposed. The security analysis shows that the preferred S-box constructed by the proposed algorithm has good cryptographic properties.

Three-state von Neumann cellular automata and 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.

SummaryThis paper presents the design and implementation of an integrated architecture for embedding security into quasi‐cyclic (QC) low‐density parity‐check (LDPC) code–based cryptographic system through a nonlinear function of low hardware complexity. Instead of using standardS‐boxes for implementation of nonlinear function, this paper considers a method on the basis of maximum length cellular automata (CA), so that enhanced security can be achieved with simple hardware structure. The proposed system adopts a lightweight random bit stream generator on the basis of linear feedback shift register (LFSR) for generating random error vectors, so that a large number of vectors with very good cryptographic properties can be made available with low hardware cost. Different permutation patterns generated for different message blocks help to provide good degrees of freedom for tuning security with reasonable key size. The hardware architecture for the proposed system is developed and validated through implementation on Xilinx Spartan 3S500E. Analytical and synthesis results show that the proposed scheme is lightweight and offers very high security through continuously changing parameters, thus making it highly suitable for resource‐constrained applications.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

We recently formulated an approach to representing structures in cellular automata (CA) spaces, and the rules that govern cell state changes, that is amenable to manipulation by genetic programming (GP). Using this approach, it is possible to efficiently generate self-replicating configurations for fairly arbitrary initial structures. Here, we investigate the properties of self-replicating systems produced using GP in this fashion as the initial configuration's size, shape, symmetry, allowable states, and other factors are systematically varied. We find that the number of GP generations, computation time, and number of resulting rules required by an arbitrary structure to self-replicate are positively and jointly correlated with the number of components, configuration shape, and allowable states in the initial configuration, but inversely correlated with the presence of repeated components, repeated sub-structures, and/or symmetric sub-structures. We conclude that GP can be used as a "replicator factory" to produce a wide range of self-replicating CA configurations, and that the properties of the resulting replicators can be predicted in part a priori. The rules controlling self-replication that are created by GP generally differ from those created manually in past CA studies.

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.