# [Resource Topic] 2022/664: The $c-$differential uniformity and boomerang uniformity of three classes of permutation polynomials over $\mathbb{F}_{2^n}$

Welcome to the resource topic for 2022/664

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

Authors: Qian Liu, Zhiwei Huang, Jianrui Xie, Ximeng Liu, and Jian Zou

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 \mathbb{F}_{2^n}, we determine the c-differential uniformity and boomerang uniformity of these permutation polynomials: (1) f_1(x)=x+\mathrm{Tr}_1^n(x^{2^{k+1}+1}+x^3+x+ux), where n=2k+1, u\in\mathbb{F}_{2^n} with \mathrm{Tr}_1^n(u)=1; (2) f_2(x)=x+\mathrm{Tr}_1^n(x^{{2^k}+3}+(x+1)^{2^k+3}), where n=2k+1; (3) f_3(x)=x^{-1}+\mathrm{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\in\mathbb{F}_{2^n}\backslash \{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).

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