Two classes of permutation trinomials with Niho exponents

Abstract

In this paper, we consider two classes of permutation trinomials with Niho-type exponents over the finite field F22m, where m is a positive integer. We transform the problem into investigating on some quartic equations (2k-th degree equations) over the subfield F2m in the first class (second class, respectively). We show that these equations have no solutions in F2m. Some sufficient conditions are established to characterize the coefficients in the two classes of permutation polynomials. The numerical result suggests that the sufficient conditions on the coefficients for the case of m odd in the first class, and in the second class, are also necessary.