The second-order zero differential spectra of some APN and other maps over finite fields

The Feistel Boomerang Connectivity Table and the related notion of [Formula: see text]-Boomerang uniformity (also known as the second-order zero differential uniformity) have been recently introduced by [H. Boukerrou, P. Huynh, V. Lallemand, B. Mandal and M. Minier, On the Feistel counterpart of the boomerang connectivity table, IACR Trans. Symmetric Cryptol. 1 (2020) 331–362]. In the same paper, a characterization of almost perfect nonlinear (APN) functions 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.