Two types of permutation polynomials with special forms

TL;DR

This paper introduces four new infinite classes of permutation trinomials over finite fields, explores their relationships with other permutation polynomial forms, and derives additional classes without restrictions on parameters.

Contribution

It proposes four families of permutation trinomials with special forms and establishes their connections to other permutation polynomial classes, expanding the known classes over finite fields.

Findings

Four new infinite classes of permutation trinomials are proposed.

Relationships between different forms of permutation polynomials are established.

Additional permutation trinomials are derived without restrictions on parameters.

Abstract

Let $q$ be a power of a prime and $\mathbb{F}_q$ be a finite field with $q$ elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form $cx-x^s + x^{qs}$ over $\mathbb{F}_{q^2}$, and investigate the relationship between this type of permutation polynomials with that of the form $(x^q-x+\delta)^s+cx$. Based on this relation, many classes of permutation trinomials having the form $(x^q-x+\delta)^s+cx$ without restriction on $\delta$ over $\mathbb{F}_{q^2}$ are derived from known permutation trinomials having the form $cx-x^s + x^{qs}$.