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

• URL: http://arxiv.org/abs/2310.13775v1
• Date: Fri, 20 Oct 2023 19:21:30 GMT
• Title: The second-order zero differential spectra of some APN and other maps over finite fields
• Authors: Kirpa Garg, Sartaj Ul Hasan, Constanza Riera, Pantelimon Stanica,
• Abstract summary: This paper presents a characterization of almost perfect nonlinear functions (APN) over fields of even characteristic in terms of second-order zero differential uniformity. We 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.
• Score: 5.537294943912028
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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