In this paper, we study the value distributions of perfect nonlinear
functions, i.e., we investigate the sizes of image and preimage sets. Using
purely combinatorial tools, we develop a framework that deals with perfect
nonlinear functions in the most general setting, generalizing several results
that were achieved under specific constraints. For the particularly interesting
elementary abelian case, we derive several new strong conditions and
classification results on the value distributions. Moreover, we show that most
of the classical constructions of perfect nonlinear functions have very
specific value distributions, in the sense that they are almost balanced.
Consequently, we completely determine the possible value distributions of
vectorial Boolean bent functions with output dimension at most 4. Finally,
using the discrete Fourier transform, we show that in some cases value
distributions can be used to determine whether a given function is perfect
nonlinear, or to decide whether given perfect nonlinear functions are
equivalent.Comment: 28 pages. minor revisions of the previous version. The paper is now
identical to the published version, outside of formattin