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)} $f(x)=\sum_{i=0}^{m-1}x_ix_{i+m} + \gamma(x_0+x_m,\cdots, x_{m-1}+x_{2m-1})$, {\rm (ii)} $f_t(x)= \sum_{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_{i}x_{i+t})+ \sum_{i=0}^{m-1}x_ix_{i+m}+ \gamma(x_0+x_m,\cdots, x_{m-1}+x_{2m-1})$, \noindent where $n=2m$, $\gamma(X_0,X_1,\cdots, X_{m-1})$ 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 $m$.