Abstract
Feistel Boomerang Connectivity Table (FBCT) is an important cryptanalytic technique on analysing the resistance of the Feistel network-based ciphers to power attacks such as differential and boomerang attacks. Moreover, the coefficients of FBCT are closely related to the second-order zero differential spectra of the function F(x) over the finite fields with even characteristic and the Feistel boomerang uniformity is the second-order zero differential uniformity of F(x). In this paper, by computing the number of solutions of specific equations over finite fields, we determine explicitly the second-order zero differential spectra of power functions x^2^m+3 and x^2^m+5 with m>2 being a positive integer over finite field with even characteristic, and x^p^k+1 with integer k≥1 over finite field with odd characteristic p. It is worth noting that x^2^m+3 is a permutation over F_2ⁿ and only when m is odd, x^2^m+5 is a permutation over F_2ⁿ, where integer n=2m. As a byproduct, we find F(x)=x⁴ is a PN and second-order zero differentially 0-uniform function over F_3ⁿ with odd n. The computation of these entries and the cardinalities in each table aimed to facilitate the analysis of differential and boomerang cryptanalysis of S-boxes when studying distinguishers and trails.