Fourier transforms and bent functions on finite groups

Abstract

Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis ${\widehat{G}}$, consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on ${\widehat{G}}$, and obtain characterizations of a bent function by its Fourier transforms (as functions on ${\widehat{G}}$). For a function f from G to another finite group, we define a dual function ${\widetilde{f}}$ on ${\widehat{G}}$, and characterize the nonlinearity of f by its dual function ${\widetilde{f}}$. Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.