Special values of Kloosterman sums and binomial bent functions

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• 作者：Chunming Tang^a ; ^{tangchunmingmath@163.com" class="auth_mail" title="E-mail the corresponding author} ; Yanfeng Qi^b
• 关键词：06E75 ; 94A60 ; 11T23
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：41
• 期：Complete
• 页码：113-131
• 全文大小：388 K

文摘

Let p≥7$p\ge 7$ and q=p^m$q=p^m$. $K_q(a)={\sum }_{x\in {\mathbb{F}}_{p^m}}{\zeta }^{{\mathrm{Tr}}_1^m(x^{p^m-2}+ax)}$ is the Kloosterman sum of a   on F_p^m${\mathbb{F}}_{p^m}$, where $\zeta =e^{\frac{2\pi \sqrt{-1}}{p}}$. The value $1-\frac{2}{\zeta +{\zeta }^{-1}}$ of K_q(a)$K_q(a)$ and its conjugate have close relationship with a class of binomial functions 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$p=11$, for any a  , $K_q(a)\ne 1-\frac{2}{\zeta +{\zeta }^{-1}}$. And for p≥13$p\ge 13$, if a∈F_p^s$a\in {\mathbb{F}}_{p^s}$ and s=gcd(2,m)$s=\gcd (2,m)$, $K_q(a)\ne 1-\frac{2}{\zeta +{\zeta }^{-1}}$. In application, these results explain that some class of binomial regular bent functions does not exist.