The differential uniformity of three classes of power functions

Power functions with low differential uniformity can be applied to cryptography, sequence designs, coding theory, and combinatorial designs. In this paper, we investigate three classes of power functions with low differential uniformity $ \Delta _{F} $ over the finite field $ {\mathbb{F}}_{p^{n}} $, where $ p $ is an odd prime and $ n $ is a positive integer. We first examine the differential uniformity $ \Delta _{F_{1}} $ of $ F_{1}(x) = x^{p^{n}-4} $ over $ {\mathbb{F}}_{p^{n}} $, which is a permutation if $ 3 \not | (p^{n}-1) $. The results show that $ F_{1} $ is a locally-APN permutation when $ p = 3 $. Then, we prove that $ 5\leq\Delta _{F_{1}}\leq6 $ when $ p = 5 $ and $ \Delta _{F_{1}}\leq8 $ for $ p>5 $. Second, by determining the number of solutions to specific equations over finite fields, we obtain that $ F_{2}(x) = x^{\frac{3^{n+1}-5}{4}} $ over $ {\mathbb{F}}_{3^{n}} $ is a permutation with $ \Delta _{F_{2}}\leq6 $, where $ n $ is odd. Finally, we prove that $ F_{3}(x) = x^{\frac{p^{n}+7}{2}} $ over $ {\mathbb{F}}_{p^{n}} $ satisfies $ \Delta _{F_{3}}\leq7 $ for $ p^n\equiv1\ ({\rm{mod}}\ 4) $.