Semigroup and Metric Characteristics of Locally Primitive Matrices and Digraphs

Abstract

The notion of local primitivity for a quadratic 0, 1-matrix of size n × 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 nonidempotentmatrices 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 Γ: 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 Γ. 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.