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

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 F2n, we determine the c-differential uniformity and boomerang uniformity of these permutation polynomials: (1) f1(x)=x+Tr1n(x2k+1+1+x3+x+ux), where n=2k+1, u∈F2n with Tr1n(u)=1; (2) f2(x)=x+Tr1n(x2k+3+(x+1)2k+3), where n=2k+1; (3) f3(x)=x−1+Tr1n((x−1+1)d+x−d), where n is even and d is a positive integer. The results show that the involutions f1(x) and f2(x) are APcN functions for c∈F2n﹨{0,1}. Moreover, the boomerang uniformity of f1(x) and f2(x) can attain 2n. Furthermore, we generalize some previous works and derive the upper bounds on the c-differential uniformity and boomerang uniformity of f3(x).