It was shown by Boukerrou et al.~\cite{Bouk} [IACR Trans. Symmetric Cryptol. 1 2020, 331--362] that the $F$-boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear functions is~$0$ on $\F_{p^n}$ ($p$ prime) and the one of almost perfect nonlinear functions on $\F_{2^n}$ is~$0$. It is natural to inquire what happens with APN or other low differential uniform functions in even and odd characteristics. Here, we explicitly determine the second-order zero differential spectra of several maps with low differential uniformity. In particular, we compute the second-order zero differential spectra for some almost perfect nonlinear (APN) functions, pushing further the study started in Boukerrou et al.~\cite{Bouk} and continued in Li et al. \cite{LYT} [Cryptogr. Commun. 14.3 (2022), 653--662], and it turns out that our considered functions also have low second-order zero differential uniformity.

翻译：Boukerrou等人~\cite{Bouk} [IACR Trans. Symmetric Cryptol. 1 2020, 331--362] 已证明，完美非线性函数在$\F_{p^n}$（$p$为素数）上的$F$-弹跳均匀度（在偶特征下等价于二阶零微分均匀度）为~$0$，而几乎完美非线性函数在$\F_{2^n}$上的该值为~$0$。这自然引发了对APN函数及偶/奇特征下其他低微分均匀函数的类似性质探究。本文精确确定了若干低微分均匀映射的二阶零微分谱。特别地，我们计算了部分几乎完美非线性（APN）函数的二阶零微分谱，进一步推进了Boukerrou等人~\cite{Bouk} 开创、Li等人 \cite{LYT} [Cryptogr. Commun. 14.3 (2022), 653--662] 延续的研究，结果表明所考虑的函数控具较低的微分均匀度阶零微分均匀度。