The Feistel Boomerang Connectivity Table and the related notion of $F$-Boomerang uniformity (also known as the second-order zero differential uniformity) has been recently introduced by Boukerrou et al.~\cite{Bouk}. These tools shall provide a major impetus in the analysis of the security of the Feistel network-based ciphers. In the same paper, a characterization of almost perfect nonlinear functions (APN) over fields of even characteristic in terms of second-order zero differential uniformity was also given. Here, we find a sufficient condition for an odd or even function over fields of odd characteristic to be an APN function, in terms of second-order zero differential uniformity. Moreover, we compute the second-order zero differential spectra of several APN or other low differential uniform functions, and show that our considered functions also have low second-order zero differential uniformity, though it may vary widely, unlike the case for even characteristic when it is always zero.

翻译：Feistel Boomerang连接表及其相关概念$F$-Boomerang一致性（亦称二阶零微分一致性）近期由Boukerrou等人提出~\cite{Bouk}。这些工具将为基于Feistel网络的密码体制安全性分析提供重要推动力。在同一论文中，作者还给出了偶特征域上几乎完全非线性函数（APN）在二阶零微分一致性意义下的刻画。本文我们基于二阶零微分一致性，给出了奇特征域上奇函数或偶函数成为APN函数的充分条件。此外，我们计算了若干APN函数及其他低微分一致函数的二阶零微分谱，结果表明所考虑的函数同样具有较低的阶零微分一致性——尽管该数值可能存在较大差异，这与偶特征情形下恒为零的情况不同。