Semigroup and metric characteristics of locally primitive matrices and graphs
The notion of local primitivity for a quadratic 0,1-matrix of size n x n is extended to any part of the matrix which need not be a rectangular submatrix. A similar generalization is carried out for any set B of pairs of initial and final vertices of the paths in an n-vertex digraph, B  {(i, j) : 1 ≤ i, j ≤ n}. We establish the relationship between the local B-exponent of a matrix (digraph) and its characteristics such as the cyclic depth and period, the number of nonprimitive matrices, and the number of nonidempotent matrices in the multiplicative semigroup of all quadratic 0, 1-matrices of order n, etc. We obtain a criterion of B-primitivity and an upper bound for the B-exponent. We also introduce some new metric characteristics for a locally primitive digraph Г : the k, r-exporadius, the k, r-expocenter, where 1 ≤ k, r ≤ n, and the matex which is defined as the matrix of order n of all local exponents in the digraph Г. An example of computation of the matex is given for the n-vertex Wielandt digraph. Using the introduced characteristics, we propose an idea for algorithmically constructing realizable s-boxes (elements of round functions of block ciphers) with a relatively wide range of sizes. Tab. 2, illustr. 1, bibliogr. 13.