LAUSR

Highly nonlinear functions (bent functions, perfect nonlinear functions, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in numerous papers. As a generalization of bent functions on finite groups, bent functions on group actions have been studied in quite a few papers. In this paper, we study Fourier transforms and bent functions on finite nonabelian group actions. Let G be a finite (nonabelian) group acting on a finite set X. Our goal is to develop a Fourier analysis on X and study the bentness of complex valued functions on X via their Fourier transforms. We will first construct a G-dual set $$\widehat{X}$$ , which is a special basis of the G-space of all complex valued functions on X. This G-dual set $${\widehat{X}}$$ plays a role similar to that of $${\widehat{G}}$$ . Then, for a complex-valued function f on X, we define its Fourier transform $${\widehat{f}}$$ as a function on $${\widehat{X}}$$ and characterize the bentness of f by its Fourier transform $${\widehat{f}}$$ . Some known and new results about bent functions on finite groups will be obtained as direct consequences. We will also discuss the constructions of bent functions and construct examples of bent functions on group actions of the dihedral group of order 6.