Determination of a Type of Permutation Trinomials over Finite Fields

Abstract

Let $f=a\mathtt{x}+b{\mathtt{x}}^q+{\mathtt{x}}^{2q-1}\in {\mathbb{F}}_q[\mathtt{x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of ${\mathbb{F}}_{q^2}$. This result allows us to solve a related problem. Let $g_{n,q}\in {\mathbb{F}}_p[\mathtt{x}]$ ($n\ge 0$, $p=\text{char}{\mathbb{F}}_q$) be the polynomial defined by the functional equation ${\sum }_{c\in {\mathbb{F}}_q}(\mathtt{x}+c{)}^n=g_{n,q}({\mathtt{x}}^q-\mathtt{x})$. We determine all $n$ of the form $n=q^{\alpha }-q^{\beta }-1$, $\alpha >\beta \ge 0$, for which $g_{n,q}$ is a permutation polynomial of ${\mathbb{F}}_{q^2}$.

28 pages

References in corpus (3)

• A Class of Binomial Permutation Polynomials
• Permutation polynomials on F_q induced from bijective Redei functions on subgroups of the multiplicative group of F_q
• A Class of Permutation Trinomials over Finite Fields

Cited by in corpus (2)

• A note regarding permutation binomials over finite fields
• Permutation Polynomials of ${\mathbb{F}}_{q^2}$ of the form $a\mathtt{X}+{\mathtt{X}}^{r(q-1)+1}$