Two types of permutation polynomials with special forms

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 $cx-x^s+x^{qs}$.