2D Triangular von Neumann Cellular Automata with Periodic Boundary

Cellular automata (CA) theory is a very rich and useful model of a discrete dynamical system that focuses on their local information relying on the neighboring cells to produce CA global behaviors. Although the main structure of CA is a discrete special model, the global behaviors at many iterative times and on big scales can be close to nearly a continuous system. The mathematical points of the basic model imply the computable values of the mathematical structure of CA. After modeling the CA structure, an important problem is to be able to move forwards and backwards on CA to understand their behaviors in more elegant ways. This happens in the possible case if CA is a reversible one. In this paper, we investigate the structure and the reversibility cases of two-dimensional (2D) finite, linear, and triangular von Neumann CA with periodic boundary case. It is considered on ternary field [Formula: see text] (i.e. 3-state). We obtain the transition rule matrices for each special case. It is known that the reversibility cases of 2D CA is generally a very challenging problem. For given special triangular information (transition) rule matrices, we prove which triangular linear 2D von Neumann CA is reversible or not. In other words, the reversibility problem of 2D triangular, linear von Neumann CA with periodic boundary is resolved completely over ternary field. However, the general transition rule matrices are also presented to establish the reversibility cases of these special 3-states CA. Since the main CA structures are sufficiently simple to investigate in mathematical ways and also very complex for obtaining chaotic models, we believe that these new types of CA can be found in many different real life applications in special cases e.g. mathematical modeling, theoretical biology and chemistry, DNA research, image science, textile design, etc. in the near future.

Structure and Reversibility of 2D von Neumann Cellular Automata Over Triangular Lattice

Even though the fundamental main structure of cellular automata (CA) is a discrete special model, the global behaviors at many iterative times and on big scales could be a close, nearly a continuous, model system. CA theory is a very rich and useful phenomena of dynamical model that focuses on the local information being relayed to the neighboring cells to produce CA global behaviors. The mathematical points of the basic model imply the computable values of the mathematical structure of CA. After modeling the CA structure, an important problem is to be able to move forwards and backwards on CA to understand their behaviors in more elegant ways. A possible case is when CA is to be a reversible one. In this paper, we investigate the structure and the reversibility of two-dimensional (2D) finite, linear, triangular von Neumann CA with null boundary case. It is considered on ternary field [Formula: see text] (i.e. 3-state). We obtain their transition rule matrices for each special case. For given special triangular information (transition) rule matrices, we prove which triangular linear 2D von Neumann CAs are reversible or not. It is known that the reversibility cases of 2D CA are generally a much challenged problem. In the present study, the reversibility problem of 2D triangular, linear von Neumann CA with null boundary is resolved completely over ternary field. As far as we know, there is no structure and reversibility study of von Neumann 2D linear CA on triangular lattice in the literature. Due to the main CA structures being sufficiently simple to investigate in mathematical ways, and also very complex to obtain in chaotic systems, it is believed that the present construction can be applied to many areas related to these CA using any other transition rules.

The Transition Rules of 2D Linear Cellular Automata Over Ternary Field and Self-Replicating Patterns

In this paper we start with two-dimensional (2D) linear cellular automata (CA) in relation with basic mathematical structure. We investigate uniform linear 2D CA over ternary field, i.e. ℤ3. Present work is related to theoretical and imaginary investigations of 2D linear CA. Even though the basic construction of a CA is a discrete model, its macroscopic level behavior at large times and on large scales could be a close approximation to a continuous system. Considering some statistical properties, someone may also study geometrical aspects of patterns generated by cellular automaton evolution. After iteratively applying the linear rules, CA have been shown capable of producing interesting complex behaviors. Some examples of CA produce remarkably regular behavior on finite configurations. Using some simple initial configurations, the produced pattern can be self-replicating regarding some linear rules. Here we deal with the theory 2D uniform periodic, adiabatic and reflexive boundary CA (2D PB, AB and RB) over the ternary field ℤ3and the applications of image processing for patterns generation. From the visual appearance of the patterns, it is seen that some rules display sensitive dependence on boundary conditions and their rule numbers.

Achieving Identity-Based Cryptography in a Personal Digital Assistant Device

Continuous technological advances have allowed that mobile devices, such as Personal Digital Assistants (PDAs), can executesophisticated applications that more often than not must be equipped with a layer of security that should include theconfidentiality and the authentication services within its repertory. Nevertheless, when compared against front-end computingdevices, most PDAs are still seen as constrained devices with limited processing and storage capabilities.In order to achieve Identity-Based Cryptography (IBC), which was an open problem proposed by Adi Shamir in 1984, Bonehand Franklin presented in Crypto 2001, a solution that uses bilinear pairings as its main building block. Since then, IBC hasbecome an active area of investigation where many efficient IBC security protocols are proposed year after year. In this paper, we present a cryptographic application that allows the secure exchange of documents from a Personal Digital Assistant (PDA) that is wirelessly connected to other nodes. The architecture of our application is inspired by the traditional PGP (Pretty Good Privacy) email security protocol. Our application achieves identity-based authentication and confidentiality functionalities at the 80-bit security level through the usage of a cryptographic library that was coded in C++. Our library can perform basic primitives such as bilinear pairings defined over the binary field and the ternary field , as well as other required primitives known as map-to-point hash functions. We report the timings achieved by our application and we show that they compare well against other similar works published in the open literature.