Two infinite classes of rotation symmetric bent functions with simple representation

Abstract

In the literature, few $n$-variable rotation symmetric bent functions have been constructed. In this paper, we present two infinite classes of rotation symmetric bent functions on ${\mathbb{F}}_2^n$ of the two forms: {\rm (i)} $f(x)={\sum }_{i=0}^{m-1}{x}_i{x}_{i+m}+{\hat{I}}^3(x_0+x_m,\cdots ,x_{m-1}+x_{2m-1})$, {\rm (ii)} $f_t(x)={\sum }_{i=0}^{n-1}(x_i{x}_{i+t}{x}_{i+m}+x_i{x}_{i+t})+{\sum }_{i=0}^{m-1}{x}_i{x}_{i+m}+{\hat{I}}^3(x_0+x_m,\cdots ,x_{m-1}+x_{2m-1})$, \noindent where $n=2m$, ${\hat{I}}^3(X_0,X_1,\cdots ,X_{m-1})$ is any rotation symmetric polynomial, and $m/gcd(m,t)$ is odd. The class (i) of rotation symmetric bent functions has algebraic degree ranging from 2 to $m$ and the other class (ii) has algebraic degree ranging from 3 to $m$.