The second-order zero differential spectra of some functions over finite fields
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It was shown by Boukerrou et al. <cit.> [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 _p^n (p prime) and the one of almost perfect nonlinear functions on _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. <cit.> and continued in Li et al. <cit.> [Cryptogr. Commun. 14.3 (2022), 653–662], and it turns out that our considered functions also have low second-order zero differential uniformity.
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