Two infinite classes of rotation symmetric bent functions with simple representation

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)} $f(x)=\sum{i=0}^{{m-1}xix{i+m}} + \gamma(x0+xm,\cdots, x{m-1}+x{2m-1})$, {\rm (ii)} $ft(x)= \sum{i=0}^{{n-1}(xix{i+t}x_{i+m}} +x{i}x{i+t})+ \sum{i=0}^{m-1}xix{i+m}+ \gamma(x0+xm,\cdots, x{m-1}+x{2m-1})$, \noindent where $n=2m$, $\gamma(X0,X1,\cdots, X{m-1})$ 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 $m$.