On the second-order zero differential spectra of some power functions over finite fields

Boukerrou et al. (IACR Trans. Symm. Cryptol. 2020(1), 331-362, 2020) introduced the notion of the Feistel Boomerang Connectivity Table (FBCT), the Feistel counterpart of the Boomerang Connectivity Table (BCT), and the Feistel boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic fields). The FBCT is a crucial table for the analysis of the resistance of block ciphers to power attacks such as differential and boomerang attacks. It is worth noting that the coefficients of the FBCT are related to the second-order zero differential spectra of functions and the FBCT of functions can be extended as their second-order zero differential spectra. In this paper, by carrying out certain finer manipulations consisting of solving some specific equations over finite fields, we explicitly determine the second-order zero differential spectra of some power functions with low differential uniformity, and show that these functions also have low second-order zero differential uniformity. Our study further pushes previous investigations on second-order zero differential uniformity and Feistel boomerang uniformity for a power function F.