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Cellular automata are systems which use a rule to describe the evolution of a population in a discrete lattice, while genetic algorithms are procedures designed to find solutions to optimization problems inspired by the process of natural selection. In this paper, we introduce an original implementation of a cellular automaton whose rules use a fitness function to select for each cell the best mate to reproduce and a crossover operator to determine the resulting offspring. This new system, with a proper definition, can be both a cellular automaton and a genetic algorithm. We show that in our system the Conway's Game of Life can be easily implemented and, consequently, it is capable of universal computing. Moreover two generalizations of the Game of Life are created and also implemented with it. Finally, we use our system for studying and implementing the prisoner's dilemma and rock-paper-scissors games, showing very interesting behaviors and configurations (e.g., gliders) inside these games.

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We present a study on the use of Pell hyperbolas in cryptosystems with security based on the discrete logarithm problem. Specifically, after introducing the group's structure over generalized Pell conics (and also giving the explicit isomorphisms with the classical Pell hyperbolas), we provide a parameterization with both an algebraic and a geometrical approach. The particular parameterization that we propose appears to be useful from a cryptographic point of view because the product that arises over the set of parameters is connected to the Rédei rational functions, which can be evaluated in a fast way. Thus, we exploit these constructions for defining three different public key cryptosystems based on the ElGamal scheme. We show that the use of our parameterization allows to obtain schemes more efficient than the classical ones based on finite fields.

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