New classes of permutation binomials and permutation trinomials over finite fields

Abstract

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinational design, communication theory and so on. Permutation binomials and trinomials attract people's interest due to their simple algebraic form and additional extraordinary properties. In this paper, several new classes of permutation binomials and permutation trinomials are constructed. Some of these permutation polynomials are generalizations of known ones.

A note on permutation polynomials over finite fields

“…In the same paper, they also proposed the following two conjectures, which can be used to obtain two new classes of permutation trinomials with the form (1). More recent progress on permutation trinomials can be found in [2,6,7,8,9,10,12,13,14,15,16,20].…”

Several classes of permutation trinomials from Niho exponents

“…The construction of permutation polynomials with few terms attracts researchers' interest in recent years due to their simple form [4,10,12,16,20,21,23,24,25] and some additional nice properties, for example, their close relations with the Helleseth's −1 conjecture [7], Welch's conjecture [5] and…”

“…Currently, only a small number of specific classes of permutation binomials and trinomials are described in the literature, see the survey paper [9] and two recent papers [4,16] on permutation binomials and trinomials over finite fields. Tu et al in [20] investigated the permutation property of a class of polynomials over F 2 n of the form…”

Constructions of Involutions Over Finite Fields

“…Conversely, it is verified that r 2 ≡ 1 mod q − 1 since r ≡ −1 mod q − 1. If h(x) has no root in µ q+1 , then from (19) we have that…”