### 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.

Available format(s)
Category
Foundations
Publication info
Published elsewhere. Unknown where it was published
Contact author(s)
xwcao @ nuaa edu cn
History
Short URL
https://ia.cr/2010/039

CC BY

BibTeX

@misc{cryptoeprint:2010/039,
author = {Xiwang Cao and Lei Hu},
title = {On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields},
howpublished = {Cryptology ePrint Archive, Paper 2010/039},
year = {2010},
note = {\url{https://eprint.iacr.org/2010/039}},
url = {https://eprint.iacr.org/2010/039}
}

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