$p(x)$-Circulants over Finite Fields and Probability Methods of Their Construction

In the paper, the algebra of $p(x)$-circulants over an arbitrary finite field is studied and algorithms of random equiprobable choice of elements in the subset of all invertible $p(x)$-circulants or in the subset of all $p(x)$-circulants with given value of the determinant are constructed. The specific feature of the algorithms under consideration is the minimization of time complexity and of the number of random elements used in the course of work of the algorithms.