  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.
