A note on permutation polynomials over finite fields

Abstract

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form x + γTr q n /q (x k ) is also presented, which generalizes two examples of [10].

Constructions of Involutions Over Finite Fields

“…For β = 1, from (22) we know that k = h(1) ∈ F * q . So, g(1) = h(1) s = k (q−1) s q−1 ≡ 1 mod p. By the second congruence of (21), we have ϕ(1) = h(g(1))h(1) ≡ k 2 ≡ 1 mod p. It is easy to verify that these parameters satisfy (21). By Theorem 4.10 we know that f (x) = x + x 1313 + x 2625 + x 3937 is an involution on F 3 8 .…”

“…If k is even, then d is odd from the first congruent equality in (21). In this case, 1 2 is viewed as the inverse of 2 modulo d. The equality (22) shows that h(β)…”

“…To show that f (x) is an involution on F q 2m , by Theorem 2.2 it is sufficient to verify that for any β ∈ µ d , ϕ(β) = h(g(β))h(β) = 1. By the first congruence of (21) we know that d | (k 2 − 1) and…”

Permutation polynomials, fractional polynomials, and algebraic curves

“…We now provide alternative and shorter proofs of two results from [11], referring to two conjectures presented in [14].…”

A new class of permutation trinomials constructed from Niho exponents

“…where l is a nonnegative integer and gcd(2l + p, q − 1) = 1. In addition, when p = 5, the permutation polynomials f (x) presented in (2) are shown to be new in the sense that they are not multiplicative equivalent to the permutation polynomials of the form (1) in [21,18,29,22,9,10,4,13].…”