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

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

2010-01-25

arxiv.org/abs/1001.4305v1

Computer Science

Information Theory

18 pages

Scientific paper

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 for 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.

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