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.

在本文中，我们研究完美非线性函数的值分布，即我们研究图像和原像集的大小。使用纯粹的组合工具，我们开发了一个框架，可以在最一般的情况下处理完美的非线性函数，概括在特定约束下取得的几个结果。对于特别有趣的基本阿贝尔情况，我们得出了几个新的强条件和值分布的分类结果。此外，我们表明，大多数完美非线性函数的经典结构都具有非常特定的值分布，从某种意义上说，它们几乎是平衡的。因此，我们完全确定了输出维度最多为 4 的矢量布尔弯曲函数的可能值分布。最后，使用离散傅立叶变换，我们表明在某些情况下，值分布可以用于确定给定函数是否是完美非线性的，或者决定给定的完美非线性函数是否等价。