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.

翻译：在本文中,我们研究了完美非线性函数的值分布, 即我们调查图像和预视集的大小。 因此, 我们使用纯粹的组合工具, 开发一个框架, 在最一般的环境下处理完美的非线性函数, 概括在特定限制下取得的若干结果。 对于特别有趣的基本边际案例, 我们从价值分布中得出了几个新的强项条件和分类结果。 此外, 我们显示, 大多数完美的非线性函数的古典构造都有非常具体的值分布, 即它们几乎是平衡的。 因此, 我们完全确定了最多4个输出层面的矢量波列弯形函数可能的值分布。 最后, 我们使用离散的 Fourier变换, 我们显示, 在某些情况下, 值分布可以用来确定给定的函数是否完美非线性, 或者决定给定的完美非线性函数是否相等 。