Determination of a class of permutation trinomials in characteristic three

Let f (X) = X(1 + aX(q(q -1)) + bX(2(q -1 ))) is an element of F-q2[X], where a, b is an element of F-q2*. 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 F-q2 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 char F-q = 3, f is a PP of F-q2 if and only if (ab)(q) = a(b(q+1) - a(q+1)) and 1 - (b/a)(q+1) is a square in F-q*. (C) 2019 Elsevier Inc. All rights reserved.