A note on a class of permutation trinomials

Abstract

Let [Formula: see text] denote the finite field with [Formula: see text] elements. In this paper, we investigate the trinomial [Formula: see text] over the finite field [Formula: see text], where [Formula: see text] with [Formula: see text] being a positive integer. We prove that the trinomial [Formula: see text] permutes [Formula: see text] if and only if [Formula: see text] and [Formula: see text] is even. This work is a continuation of the previous work of Bai and Xia [A new class of permutation trinomials constructed from Niho exponents,Cryptogr. Commun. 10 (2018) 1023–1036].

Some results on permutation polynomials over finite fields

“…Conversely, one can start with any permutation rational function over F q and compose on both sides with suitable degreeone rational functions over F q 2 in order to get a rational function over F q 2 which permutes the (q + 1)-th roots of unity, then we can get the corresponding permutation polynomials over F q 2 . By using this strategy, Gupta and Rai [6] investigated the trinomial f (x) = x 4q+1 + αx 5q + x q+4 over the finite field F 5 2k , where α ∈ F * 5 k with k being a positive integer. They proved that the trinomial f (x) permutes F 5 2k if and only if α = −1 and k is even.…”

“…In Section 2, motivated by [6,16], we study the permutation property of polynomial with the form x q+1+m (ax n(q−1) + x m(q−1) + 1), where m, n are positive integers with m < n and 2n − m = p. We notice that when we take l(x) = δ(x+1)…”