Second-order Cellular Automata Research Articles

Modelling reliability of reversible circuits with 2D second-order cellular automata

The cellular automaton is a widely known model of both reversible and irreversible computations. The family of reversible second-order cellular automata considered in this work is appropriate both for construction of logic gates and analysis of damage distribution. The quantities such as formal dimension of damage patterns can be used only for rough estimation of consequences of particular faults and numerical experiments are provided for illustration of some subtleties. Such analysis demonstrates high sensitivity to errors from defects, lack of synchronization and too short intervals between signals.

Design of image cipher using life-like cellular automata and chaotic map

Abstract Recently, a number of image ciphers using cellular automata (CA) or combined with chaotic maps have been proposed. However, most of them suffer from some intrinsic drawbacks such as small rule space, low diffusion and no explicit classification of CA rules. To overcome these drawbacks, this paper presents a novel image cipher based on Life-like cellular automata and chaos. The proposed image cipher consists of two sub-processes: permutation and substitution. In the permutation, a two-dimensional Logistic-adjusted-Sine map (2D-LASM) with excellent properties is adopted to shuffle the pixel positions. In the substitution, a second-order Life-like CA with a balanced rule is employed. The balanced rules make the distribution of 0 and 1 in Life-like CA gradually be in equilibrium during the process of iteration. Second-order CA can preserve the result of CA after each iteration to obtain the reversibility. Furthermore, to resist chosen-plaintext and known-plaintext attacks, the algorithm controls the initial conditions of 2D-LASM by the key and the weighted histogram of the plain-image. Theoretical analysis and experimental results both show that the proposed scheme has prominent cryptographic performances and can resist the common attacks effectively, which is very suitable for image encryption.

A genetic strategy to design cellular automata based block ciphers

We propose an evolutionary computation approach to design a fast and secure block cipher using non-uniform second-order cellular automata. We build a flexible block ciphering model that permit the construction of a huge space of possible instances defined each one by a finite set of elementary transition rules. The constructed space is explored using a genetic algorithms strategy in order to find an optimal solution with respect to the strict avalanche criterion used as fitness measurements. The genetically designed cipher is benchmarked experimentally using conventional statistical tests, and shown to have very admissible characteristics leading to a very acceptable level of cryptographic security. Moreover, performances analysis shows that the designed cipher permit to achieve a high encryption/decryption speed, and compete many of the existing standardized ciphers.

Conserved quantities in discrete dynamics: what can be recovered from Noether’s theorem, how, and why?

The connections between symmetries and conserved quantities of a dynamical system brought to light by Noether's theorem depend in an essential way on the symplectic nature of the underlying kinematics. In the discrete dynamics realm, a rather suggestive analogy for this structure is offered by second-order cellular automata. We ask to what extent the latter systems may enjoy properties analogous to those conferred, for continuous systems, by Noether's theorem. For definiteness, as a second-order cellular automaton we use the Ising spin model with both ferromagnetic and antiferromagnetic bonds. We show that--and why--energy not only acts as a generator of the dynamics for this family of systems, but is also conserved when the dynamics is time-invariant. We then begin to explore the issue of whether, in these systems, it may hold as well that translation invariance entails momentum conservation.

Linear reversible second-order cellular automata and their first-order matrix equivalents

Linear or one-dimensional reversible second-order cellular automata, exemplified by three cases named as RCA1–3, are introduced. Displays of their evolution in discrete time steps, , from their simplest initial states and on the basis of updating rules in modulo 2 arithmetic, are presented. In these, shaded and unshaded squares denote cells whose cell variables are equal to one and zero respectively. This paper is devoted to finding general formulas for, and explicit numerical evaluations of, the weights N(t) of the states or configurations of RCA1–3, i.e. the total number of shaded cells in tth line of their displays. This is achieved by means of the replacement of RCA1–3 by the equivalent linear first-order matrix automata MCA1–3, for which the cell variables are matrices, instead of just numbers () as for RCA1–3. MCA1–3 are tractable because it has been possible to generalize to them the heavy duty methods already well-developed for ordinary first-order cellular automata like those of Wolfram's Rules 90 and 150. While the automata MCA1–3 are thought to be of genuine interest in their own right, with untapped further mathematical potential, their treatment has been applied here to expediting derivation of a large body of general and explicit results for N(t) for RCA1–3. Amongst explicit results obtained are formulas also for each of RCA1–3 for the total weight of the configurations of the first times, .

How to turn a second-order cellular automaton into a lattice gas: a new inversion scheme

In second-order cellular automata, the signals made available to the local transition function include, besides the current state of a site's neighbors, also the previous state of the site itself. Some similarities had been noted, especially in physics simulations, between certain second-order cellular automata and certain lattice gases. Here we show how to construct, for any second-order cellular automaton, a lattice gas having isomorphic functional behavior. (Paradoxically, this isomorphism of function is achieved by compromising on the isomorphism of structure: namely, the group of translation symmetries of the resulting lattice gas is coarser than that of the original cellular automaton.) The advantage of our construction is that, while invertibility in cellular automata is not directly deducible from the local map (indeed, it is in general undecidable), in the second-order case the invertibility of the cellular automaton goes hand-in-hand with that of the corresponding lattice gas—and for lattices gases invertibility is trivially decidable on the basis of the local structure. From a physical viewpoint, our construction illustrates a trade-off between two ways of achieving a “force” of a given range. In a cellular automaton, multiple copies of a signal are made and distributed in parallel to several sites, explicitly providing the desired fanout width. In a lattice gas, with our construction, one can let a single signal serially service a number of sites along an appropriate spacetime route. The latter method is better suited to a nondissipative implementation of the dynamics.

Invertible operations on a Cellular Neural Network Universal Machine

This paper is concerned with the invertibility of parallel operations definable on two-dimensional binary arrays by means of a local Boolean function. In this way, it contributes to the theory of cellular automata and may open new potentialities to cellular neural network applications. We provide a theoretically new approach to the invertibility of two-dimensional additive maps which can be used to obtain detailed information on the behaviour of these operations. Other ways to make invertible operations are also investigated, including local maps which are additive in only one of the terms, and second-order cellular automata. The discussed techniques yield a rich variety of invertible operations on infinite and/or finite two-dimensional binary arrays. When implemented on CNN Universal Chips, all of these operations, performed by TeraOPS speed, become subroutines of a 2D analogic algorithm. These operations can be used in 2D cryptography. © 1998 John Wiley & Sons, Ltd.