On Some Permutation Trinomials in Characteristic Three

In this paper, we determine the permutation properties of the polynomial x³+xq+2−x⁴q−1 over the finite field Fq2 in characteristic three. Moreover, we consider the trinomials of the form x⁴q−1 +x²q+1 ±x³. In particular, we first show that x³ +xq+2 −x⁴q−1 permutes Fq² with q = 3ᵐ if and only if m is odd. This enables us to show that the sufficient condition in 35, Theorem 4 is also necessary. Next, we prove that x⁴q−1 + x²q+1 − x³ permutes Fq² with q = 3ᵐ if and only if m is congruent to 0 modulo 4. Consequently, we prove that the sufficient condition in 20, Theorem 3. 2 is also necessary. Finally, we investigate the trinomial x⁴q−1 +x²q+1 +x³ and show that it is never a permutation polynomial of Fq² in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.