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

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.Comment: 18 page

On the existence of some specific elements in finite fields of characteristic 2

AbstractLet q be a power of 2, n be a positive integer, and let Fqn be the finite field with qn elements. In this paper, we consider the existence of some specific elements in Fqn. The main results obtained in this paper are listed as follows:(1)There is an element ξ in Fqn such that both ξ and ξ+ξ−1 are primitive elements of Fqn if q=2s, and n is an odd number no less than 13 and s>4.(2)For q=2s, and any odd n, there is an element ξ in Fqn such that ξ is a primitive normal element and ξ+ξ−1 is a primitive element of Fqn if either n|(q−1), and n⩾33, or n∤(q−1), and n⩾30, s⩾6