Cellular automata algorithms for matrix permutations

Four algorithms of matrix elements mixture based on cyclic shifts of rows and columns are described by cellular automata formalisation. One of the algorithms shows an interesting behaviour for odd matrix orders, when, as a result of permutations, the matrix undergoes±90∘ and 180∘ (reflection relative to the centre) rotations. The period N growth rate is morethan exponential. Based on the analysis of the short series n = 3, 5, . . . , 11, a hypothesis is proposed that N is equal to the lowest common multiple of the odd numbers less than 2n, i.e. N = LCM(3, 5, . . . , 2n - 1). Arguments in favour of the hypothesis are given. The dynamics of permutations are analysed using the two «metrics» introduced by the authors, which reflect the degree of mixedness. The results of this work can be used to generate pseudorandom numbers.