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 $\mathbb{F}_{2^n}$, 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 $f_3(x)$.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Published elsewhere. Finite Fields and Their Applications
 DOI
 10.1016/j.ffa.2023.102212
 Keywords
 $C$differential uniformityBoomerang uniformityPermutation polynomial
 Contact author(s)
 lqmova @ foxmail com
 History
 20230609: revised
 20220528: received
 See all versions
 Short URL
 https://ia.cr/2022/664
 License

CC BY
BibTeX
@misc{cryptoeprint:2022/664, author = {Qian Liu and Zhiwei Huang and Jianrui Xie and Ximeng Liu and Jian Zou}, title = {The $c$differential uniformity and boomerang uniformity of three classes of permutation polynomials over $\mathbb{F}_{2^n}$}, howpublished = {Cryptology ePrint Archive, Paper 2022/664}, year = {2022}, doi = {10.1016/j.ffa.2023.102212}, note = {\url{https://eprint.iacr.org/2022/664}}, url = {https://eprint.iacr.org/2022/664} }