On coefficients of powers of polynomials and their compositions over finite fields

For any given polynomial f over the finite field Fq with degree at most q−1, we associate it with a q×q matrix A(f) = (aik) consisting of coefficients of its powers (f(x)) = ∑q−1 i=0 aikx i modulo x−x for k = 0, 1, . . . , q−1. This matrix has some interesting properties such as A(g◦f) = A(f)A(g) where (g◦f)(x) = g(f(x)) is the composition of the polynomial g with the polynomial f . In particular, A(f(k)) = (A(f)) for any k-th composition f(k) of f with k ≥ 0. As a consequence, we prove that the rank of A(f) gives the cardinality of the value set of f . Moreover, if f is a permutation polynomial then the matrix associated with its inverse A(f(−1)) = A(f)−1 = PA(f)P where P is an antidiagonal permutation matrix. As an application, we study the period of a nonlinear congruential pseduorandom sequence ā = {a0, a1, a2, ...} generated by an = f(a0) with initial value a0, in terms of the order of the associated matrix. Finally we show that A(f) is diagonalizable in some extension field of Fq when f is a permutation polynomial over Fq.