The c-differential uniformity and boomerang uniformity of three classes of permutation polynomials over [formula omitted].

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 2 n , we determine the c -differential uniformity and boomerang uniformity of these permutation polynomials: (1) f 1 (x) = x + Tr 1 n (x 2 k + 1 + 1 + x 3 + x + u x) , where n = 2 k + 1 , u ∈ F 2 n with Tr 1 n (u) = 1 ; (2) f 2 (x) = x + Tr 1 n (x 2 k + 3 + (x + 1) 2 k + 3) , where n = 2 k + 1 ; (3) f 3 (x) = x − 1 + Tr 1 n ((x − 1 + 1) d + x − d) , where n is even and d is a positive integer. The results show that the involutions f 1 (x) and f 2 (x) are APcN functions for c ∈ 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). [ABSTRACT FROM AUTHOR]