Two shorter proofs on the inverse and differential spectrum of Bracken-Leander exponent

In this paper, we investigate the Bracken-Leander power function F(x) = x(22k+2k+1) over F-24k where k is an odd positive integer, and first give a much shorter proof on the binary representation of its inverse based on the Chinese Remainder Theorem. Besides, based on a known connection between the differential spectrum and Fourier spectrum of a function, we also give another shorter proof to determine the differential spectrum of F(x). These two results are solved recently with quite involved skills by Kolsch (2020) [7], Xiong and Yan (2017) [10], respectively. We hope that our work is helpful to have a better understanding of this function because of its importance in the construction of S-boxes in block ciphers. (c) 2021 Elsevier B.V. All rights reserved.