Nonlinear functions in abelian groups and relative difference sets

During the past decade, perfect, almost perfect and maximum nonlinear functions on finite fields have been thoroughly investigated. The main tool to investigate these functions is the Walsh-Hadamard transform. This is a special version of the more general discrete Fourier transform. It is the purpose of this paper to show that the main results on nonlinear functions can be easily generalized to the case of arbitrary abelian groups if the Walsh-Hadamard transform is replaced by the discrete Fourier transform. This approach has three advantages: Proofs become more transparent. The connection with (relative) difference sets becomes apparent. It yields possible generalizations to nonlinear functions on abelian groups. (C) 2003 Elsevier B.V. All rights reserved.