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：本文研究完美非线性函数的值分布，即探究其像集和原像集的大小。我们利用纯组合工具，在最一般的情形下建立了处理完美非线性函数的理论框架，推广了若干在特定约束条件下得到的已有结果。特别地，对于具有重要意义的初等阿贝尔情形，我们推导出关于值分布的多个新强条件及分类结果。此外，研究表明大多数经典完美非线性函数构造具有极为特殊的值分布——它们几乎是平衡的。基于此，我们完整确定了输出维数不超过4的向量布尔弯曲函数可能的值分布。最后，通过离散傅里叶变换，我们证明了在某些情形下值分布可用于判定给定函数是否为完美非线性函数，或判定给定的完美非线性函数是否等价。