**On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields**

*Xiwang Cao and Lei Hu*

**Abstract: **Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}}\chi^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with $a\neq 0$, $u$ be a positive integer. We obtain an estimate of the exponential sum $\sum_{c\in \mathbb{F}^*_{q^s}}\chi^{(s)}(ac^u+bc^{-1})$, where $\chi^{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided also.

**Category / Keywords: **foundations /

**Date: **received 23 Jan 2010

**Contact author: **xwcao at nuaa edu cn

**Available format(s): **PDF | BibTeX Citation

**Version: **20100126:051054 (All versions of this report)

**Short URL: **ia.cr/2010/039

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