Two classes of permutation polynomials over finite fields

Two classes of permutation polynomials over finite fields are presented. The first class is a further study of permutation polynomials of the form (xpk−x+δ)s+L(x) and the second class is a supplement of the recent work of Hou on permutation polynomials. We show the permutation properties of two polynomials in the first class and five polynomials in the second class by using their implicit or explicit piecewise function characteristic over the subsets of the finite field defined by multiplicative or additive characters of the field. Two polynomials in the first class theoretically explain two numerical observations of J. Yuan et al. in their permutation polynomial search experiment.