Nonlinear functions and difference sets on group actions

Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G,H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat{X}$. We will also investigate the existence and constructions of $G$-perfect nonlinear functions and $G$-bent functions. Several known results in [2,6,10,17] are direct consequences of our results.