Further results on permutation trinomials over finite fields with even characteristic

Permutation trinomials of the form xrh(x2m−1) over F22m are investigated in this paper, which is a further study on a recent work of Gupta and Sharma. Based on some bijections over the unit circle of F22m with order 2m+1, the two conjectures proposed by Gupta and Sharma are confirmed and several new permutation trinomials are presented.