Paper 2022/664

The $c-$differential uniformity and boomerang uniformity of three classes of permutation polynomials over ${\mathbb{F}}_{2^n}$

Abstract

Permutation polynomials with low $c$-differential uniformity and boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over $$, we determine the $$-differential uniformity and boomerang uniformity of these permutation polynomials: (1) $$, where $$, $$ with $$; (2) $$, where $$; (3) $$, where $$ is even and $$ is a positive integer. The results show that the involutions $$ and $$ are APcN functions for $$. Moreover, the boomerang uniformity of $$ and $$ can attain $$. Furthermore, we generalize some previous works and derive the upper bounds on the $$-differential uniformity and boomerang uniformity of $$.