We determine a connection between the weight of a Boolean function and the total weight of its first-order derivatives. The relationship established is used to study some cryptographic properties of Boolean functions. We establish a characterization of APN permutations in terms of the weight of the first-order derivatives of their components. We also characterize APN functions by the total weight of the second-order derivatives of their components. The total weight of the first-order and second-order derivatives for functions such as permutations, bent, partially-bent, quadratic, plateaued and balanced functions is determined.

翻译：我们确定了布尔函数的权重与其一阶导数总权重之间的联系。所建立的关系被用于研究布尔函数的一些密码学性质。我们基于分量函数的一阶导数权重给出了APN置换的刻画，同时通过分量函数的二阶导数总权重对APN函数进行了特征化。此外，还确定了置换函数、Bent函数、部分Bent函数、二次函数、平顶函数和平衡函数等类型函数的一阶与二阶导数总权重。