The differential spectrum of a power permutation
Furong Bao, Yongbo Xia, Shaoping Chen, Chunlei Li, Tor Helleseth · Advances in Mathematics of Communications · 2024
Let $ n $ be a positive integer, $ p $ an odd prime, $ d = \frac{p^n+1}{4} $ if $ p^n \equiv3 (\bmod 8) $ and $ d = \frac{p^n+1}{4}+\frac{p^n-1}{2} $ if $ p^n \equiv7 (\bmod 8) $. It was shown that the differential uniformity of the power permutation $ x^d $ over $ \mathbb{F}_{p^n} $ is no more than $ 4 $. In this paper, we further investigate the differential properties of this power permutation. By studying the differential equation of $ x^d $ and certain equation system over $ \mathbb{F}_{p^n} $, we completely determined the differential spectrum of this power permutation.