### 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 $f_3(x)$.

Available format(s)
Category
Foundations
Publication info
Preprint.
Keywords
$C$-differential uniformity Boomerang uniformity Permutation polynomial
Contact author(s)
lqmova @ foxmail com
History
2022-05-31: approved
See all versions
Short URL
https://ia.cr/2022/664

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},
note = {\url{https://eprint.iacr.org/2022/664}},
url = {https://eprint.iacr.org/2022/664}
}

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