Further explorations in the second-order zero differential spectra of power functions over finite fields

Boukerrou et al. (IACR Trans Symmetric 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). 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. In this paper, by carrying out certain finer manipulations of solving specific equations over finite fields, we explicitly determine the second-order zero differential spectra of some power functions. Our study further pushes previous investigations on second-order zero differential uniformity and Feistel boomerang uniformity for a power function F.