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**2010/039**

**Title:**

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

**Authors:**
Xiwang Cao, 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.

**ePrint:**
https://eprint.iacr.org/2010/039

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