Constructions of rotation symmetric bent functions with high algebraic degree

Abstract

Rotation symmetric (RotS) bent functions have attracted much attention due to their cryptographic significance in the last few years. However, few known constructions of RotS bent functions with degree higher than 3 are available in the literature. In this paper, we first introduce a systematic construction of $n$-variable RotS bent functions for $n=2m=4pr$, $p\ge 2$, $r\ge 1$. The new RotS bent functions possess high degree $p+1$. For a fixed $n$, the number of such functions is ${\sum }_{p|\frac{m}{2},p>1}^{}{2}^{2^{p-2}}$ and it leads to a great increase in the number of known RotS bent functions. In addition, using any $n$-variable RotS bent function with degree $p+1$ we constructed, we deduce a new class of $n$-variable RotS bent functions with degree ranging from $p+1$ to $m$.