Several classes of bent functions over finite fields

Abstract

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 $f(x)=g(x)+F(f_1(x),f_2(x),\ldots ,$ $f_{\tau }(x))$, where g(x) is a function from $\mathbb {F}_{p^{n}}$ to ${\mathbb {F}}_{p}$, $f_i(x)=\prod _{j=1}^{\kappa _i}{\text {Tr}}(u_{ij}x)$ for $1\le i\le \tau$, $\kappa _i\ge 1$ and $F(x_1,\ldots ,x_{\tau })$ is a reduced polynomial in ${\mathbb {F}}_{p}[x_1,\ldots ,x_{\tau }]$. As a consequence, we obtain a generic result on the Walsh transform of f(x) in terms of g(x) and characterize the bentness of f(x) for the cases $F(x_1,\ldots ,x_{\tau })$ without and with restrictions respectively, which enables us to generalize some earlier works and derive new bent functions from known ones. 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.