Communicated by A. Pott.
Perfect nonlinear functions between two finite abelian groups were studied by Carlet and Ding (J Complex 20:205–244, 2004) and Pott (Discret Math Appl 138:177–193, 2004), which can be regarded as a generalization of bent functions on finite abelian groups studied by Logachev et al. (Discret Math Appl 7:547–564, 1997). Poinsot (J Discret Math Sci Cryptogr 9:349–364, 2006), (Cryptogr Commun 4:1–23, 2012) extended this research to arbitrary finite groups, and characterized bent functions on finite nonabelian groups as well as perfect nonlinear functions between two arbitrary finite groups by the Fourier transforms of the related functions at irreducible unitary representations. The purpose of this paper is to study the characterizations of the bentness (perfect nonlinearity) of functions on arbitrary finite groups by the Fourier transforms of the related functions at irreducible characters. We will also give a characterization of a perfect nonlinear function by the relative pseudo-difference family.