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In this note, using rather elementary technique and the derived formula that relates the coefficients of a polynomial over a finite field and its derivative, we deduce many interesting results related to derivatives of Boolean functions and derivatives of mappings over finite fields. For instance, we easily identify several infinite classes of polynomials… (More)

In this paper we find an exact formula for the number of partitions of an element z into m parts over a finite field, i.e. we find the number of nonzero solutions of the equation x 1 + x 2 + · · · + x m = z over a finite field when the order of terms does not matter. This is equivalent to counting the number of m-multi-subsets whose sum is z. When the order… (More)

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) = (a ik) consisting of coefficients of its powers (f (x)) k = q−1 i=0 a ik x i modulo x q −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… (More)