Special values of Kloosterman sums and binomial bent functions

Abstract

Let $p\ge 7$, $q=p^m$. $K_q(a)={\sum }_{x\in {\mathbb{F}}_{p^m}}{\zeta }^{{\mathrm{Tr}}_1^m(x^{p^m-2}+ax)}$ is the Kloosterman sum of $a$ on ${\mathbb{F}}_{p^m}$, where $\zeta =e^{\frac{2\pi \sqrt{-1}}{p}}$. The value $1-\frac{2}{\zeta +{\zeta }^{-1}}$ of $K_q(a)$ and its conjugate have close relationship with a class of binomial function with Dillon exponent. This paper first presents some necessary conditions for $a$ such that $K_q(a)=1-\frac{2}{\zeta +{\zeta }^{-1}}$. Further, we prove that if $p=11$, for any $a$, $K_q(a)\ne 1-\frac{2}{\zeta +{\zeta }^{-1}}$. And for $p\ge 13$, if $a\in {\mathbb{F}}_{p^s}$ and $s=\gcd (2,m)$, $K_q(a)\ne 1-\frac{2}{\zeta +{\zeta }^{-1}}$. In application, these results explains some class of binomial regular bent functions does not exits.