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：本文研究完全非线性函数的值分布，即探讨像集与原像集的大小。利用纯组合工具，我们建立了一个在最一般框架下处理完全非线性函数的理论，推广了在特定约束条件下获得的若干结果。对于特别重要的初等阿贝尔情形，我们推导出关于值分布的若干新的强条件与分类结果。进一步地，我们证明大多数经典构造的完全非线性函数具有极为特殊的值分布，即近乎平衡。由此，我们完全确定了输出维度不超过4的向量布尔弯曲函数的可能值分布。最后，利用离散傅里叶变换，我们证明在某些情况下值分布可用于判定给定函数是否为完全非线性函数，或判断给定的完全非线性函数是否等价。