Permutation trinomials over $\mathbb{F}_{q^3}$

TL;DR

This paper investigates specific classes of permutation polynomials over finite fields of the form _{q^3}, providing conditions on coefficients for permutation behavior and estimating the number of such coefficient pairs.

Contribution

It characterizes when four classes of polynomials over _{q^3} are permutation polynomials and estimates the quantity of coefficient pairs that produce permutations.

Findings

Conditions on (A,B) for permutation property are established.

Lower bounds on the number of permutation pairs are provided.

Four classes of polynomials are analyzed in detail.

Abstract

We consider four classes of polynomials over the fields $\mathbb{F}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+Ax^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+Ax^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+Ax^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+Ax^{q}-Bx$, where $A,B \in \mathbb{F}_q$. We determine conditions on the pairs $(A,B)$ and we give lower bounds on the number of pairs $(A,B)$ for which these polynomials permute $\mathbb{F}_{q^3}$.