Several classes of bent functions over finite fields

Abstract

Let ${\mathbb{F}}_{p^n}$ be the finite field with $p^n$ elements and $\mathrm{Tr}(\cdot )$ be the trace function from ${\mathbb{F}}_{p^n}$ to ${\mathbb{F}}_p$, where $p$ is a prime and $n$ is an integer. Inspired by the works of Mesnager (IEEE Trans. Inf. Theory 60(7): 4397-4407, 2014) and Tang et al. (IEEE Trans. Inf. Theory 63(10): 6149-6157, 2017), we study a class of bent functions of the form $f(x)=g(x)+F(\mathrm{Tr}(u_{1}x),\mathrm{Tr}(u_{2}x),\cdots ,\mathrm{Tr}(u_{\tau }x))$, where $g(x)$ is a function from ${\mathbb{F}}_{p^n}$ to ${\mathbb{F}}_p$, $\tau \ge 2$ is an integer, $F(x_1,\cdots ,x_n)$ is a reduced polynomial in ${\mathbb{F}}_p[x_1,\cdots ,x_n]$ and $u_i\in {\mathbb{F}}_{p^n}^{\ast }$ for $1\le i\le \tau$. As a consequence, we obtain a generic result on the Walsh transform of $f(x)$ and characterize the bentness of $f(x)$ when $g(x)$ is bent for $p=2$ and $p>2$ respectively. Our results generalize some earlier works. In addition, we study the construction of bent functions $f(x)$ when $g(x)$ is not bent for the first time and present a class of bent functions from non-bent Gold functions.