Bent Functions on Finite Nonabelian Groups and Relative Difference Sets

Abstract

It is well known that the perfect nonlinearity of a function between finite groups $G$ and $H$ can be characterized by its graph in terms of relative difference set in the direct product $G\times H$ (cf. [4]). Let $T$ be the infinite set of complex roots of unity. A $T$-valued function $f$ on an arbitrary finite group $G$ is associated with a finite cyclic subgroup $T_f$ in the multiplicative group of nonzero complex numbers. For a bent function $f$ on $G$ in general, its graph is not a relative difference set in the direct product $G\times T_f$. In this paper, we investigate the necessary and sufficient conditions under which the graph of a bent function $f$ on $G$ is a relative difference set in $G\times T_f$. Cyclotomic fields and their integral bases play an important role in our discussions.