Modelling Nonlinear Sequence Generators in terms of Linear Cellular Automata

In this work, a wide family of LFSR-based sequence generators, the so-called Clock-Controlled Shrinking Generators (CCSGs), has been analyzed and identified with a subset of linear Cellular Automata (CA). In fact, a pair of linear models describing the behavior of the CCSGs can be derived. The algorithm that converts a given CCSG into a CA-based linear model is very simple and can be applied to CCSGs in a range of practical interest. The linearity of these cellular models can be advantageously used in two different ways: (a) for the analysis and/or cryptanalysis of the CCSGs and (b) for the reconstruction of the output sequence obtained from this kind of generators.Comment: 15 pages, 0 figure

Linear solutions for cryptographic nonlinear sequence generators

This letter shows that linear Cellular Automata based on rules 90/150 generate all the solutions of linear difference equations with binary constant coefficients. Some of these solutions are pseudo-random noise sequences with application in cryptography: the sequences generated by the class of shrinking generators. Consequently, this contribution show that shrinking generators do not provide enough guarantees to be used for encryption purposes. Furthermore, the linearization is achieved through a simple algorithm about which a full description is provided

On the Parity Problem in One-Dimensional Cellular Automata

We consider the parity problem in one-dimensional, binary, circular cellular automata: if the initial configuration contains an odd number of 1s, the lattice should converge to all 1s; otherwise, it should converge to all 0s. It is easy to see that the problem is ill-defined for even-sized lattices (which, by definition, would never be able to converge to 1). We then consider only odd lattices. We are interested in determining the minimal neighbourhood that allows the problem to be solvable for any initial configuration. On the one hand, we show that radius 2 is not sufficient, proving that there exists no radius 2 rule that can possibly solve the parity problem from arbitrary initial configurations. On the other hand, we design a radius 4 rule that converges correctly for any initial configuration and we formally prove its correctness. Whether or not there exists a radius 3 rule that solves the parity problem remains an open problem.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Stable Multi-Level Monotonic Eroders

Eroders are monotonic cellular automata with a linearly ordered state set that eventually wipe out any finite island of nonzero states. One-dimensional eroders were studied by Gal'perin in the 1970s, who presented a simple combinatorial characterization of the class. The multi-dimensional case has been studied by Toom and others, but no such characterization has been found. We prove a similar characterization for those one-dimensional monotonic cellular automata that are eroders even in the presence of random noise.Comment: 32 pages, 9 figure

Critical droplets in Metastable States of Probabilistic Cellular Automata

We consider the problem of metastability in a probabilistic cellular automaton (PCA) with a parallel updating rule which is reversible with respect to a Gibbs measure. The dynamical rules contain two parameters $\beta$ and $h$ which resemble, but are not identical to, the inverse temperature and external magnetic field in a ferromagnetic Ising model; in particular, the phase diagram of the system has two stable phases when $\beta$ is large enough and $h$ is zero, and a unique phase when $h$ is nonzero. When the system evolves, at small positive values of $h$, from an initial state with all spins down, the PCA dynamics give rise to a transition from a metastable to a stable phase when a droplet of the favored $+$ phase inside the metastable $-$ phase reaches a critical size. We give heuristic arguments to estimate the critical size in the limit of zero ``temperature'' ($\beta\to\infty$), as well as estimates of the time required for the formation of such a droplet in a finite system. Monte Carlo simulations give results in good agreement with the theoretical predictions.Comment: 5 LaTeX picture

Cellular automata segmentation of brain tumors on post contrast MR images

In this paper, we re-examine the cellular automata(CA) al- gorithm to show that the result of its state evolution converges to that of the shortest path algorithm. We proposed a complete tumor segmenta- tion method on post contrast T1 MR images, which standardizes the VOI and seed selection, uses CA transition rules adapted to the problem and evolves a level set surface on CA states to impose spatial smoothness. Val- idation studies on 13 clinical and 5 synthetic brain tumors demonstrated the proposed algorithm outperforms graph cut and grow cut algorithms in all cases with a lower sensitivity to initialization and tumor type