Determination of a class of permutation trinomials in characteristic three

Abstract

Let f(X)=X(1+aXq(q−1)+bX2(q−1))∈Fq2[X], where a,b∈Fq2⁎. In a series of recent papers by several authors, sufficient conditions on a and b were found for f to be a permutation polynomial (PP) of Fq2 and, in characteristic 2, the sufficient conditions were shown to be necessary. In the present paper, we confirm that in characteristic 3, the sufficient conditions are also necessary. More precisely, we show that when charFq=3, f is a PP of Fq2 if and only if (ab)q=a(bq+1−aq+1) and 1−(b/a)q+1 is a square in Fq⁎.