The c-differential uniformity and boomerang uniformity of three classes of permutation polynomials over F₂ⁿ

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 F-2n, we determine the c-differential uniformity and boomerang uniformity of these permutation polynomials: (1) f1(x) = x + Tr-1(n)( x(2k+1)+1+ x(3)+ x + ux), where n = 2k+ 1, u is an element of F-2n with Tr-1(n)(u) = 1; (2) f(2)(x) = x + Tr-1(n)( x(2k+3)+( x + 1)(2k)+3), where n = 2k+ 1; (3) f(3)(x) = x(-1)+ Tr-1(n)(( x(-1)+ 1)(d)+ x(-d)), where nis even and dis a positive integer. The results show that the involutions f(1)(x) and f(2)(x) are APcN functions for c is an element of F(2)n\{0, 1}. Moreover, the boomerang uniformity of f(1)(x) and f(2)(x) can attain 2(n). Furthermore, we generalize some previous works and derive the upper bounds on the c-differential uniformity and boomerang uniformity of f(3)(x). (c) 2023 Elsevier Inc. All rights reserved.