Three-dimensional cellular automata and VLSI applications

Finite, three-dimensional (3-D), N*(N*N) cellular automata with boundary conditions are presented and discussed. It is shown that, depending on their local rule and the dimension N, these cellular automata exhibit group or semigroup algebraic structures similar to those in the one and two-dimensional (2-D) cases. The algebraic properties of these 3-D cellular automata are exploited in the implementation of integer modulo arithmetic units. Lower bounds on area A, time T, energy AT and AT² complexity metrics of 3-D cellular automata-based modulo arithmetic units are also presented.