Two new classes of permutation trinomials over [formula omitted] with odd characteristic

Abstract

Let q be an odd prime power and Fq3 be the finite field with q3 elements. In this paper, we propose two classes of permutation trinomials of Fq3 for an arbitrary odd characteristic based on the multivariate method and some subtle manipulation of solving equations with low degrees over finite fields. Moreover, we demonstrate that these two classes of permutation trinomials are QM inequivalent to all known permutation polynomials over Fq3. To the best of our knowledge, this paper is the first to study the construction of nonlinearized permutation trinomials of Fq3 with at least one coefficient lying in Fq3﹨Fq.