Classification of some quadrinomials over finite fields of odd characteristic

Abstract

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial f(x)=x3+axq+2+bx2q+1+cx3q, where a,b,c∈Fq⁎, is a permutation quadrinomial of Fq2 over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where char(Fq)=2 and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x3+axq+2+bx2q+1+cx3q, where char(Fq)=3,5 and a,b,c∈Fq⁎ and proposed some new classes of permutation quadrinomials of Fq2.In particular, in this paper we classify all permutation polynomials of Fq2 of the form f(x)=x3+axq+2+bx2q+1+cx3q, where a,b,c∈Fq⁎, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.