**Permutation Polynomials and Their Differential Properties over Residue Class Rings**

*Yuyin Yu and Mingsheng Wang*

**Abstract: **This paper mainly focuses on permutation polynomials over the residue class ring $\mathbb{Z}_{N}$, where $N>3$ is composite. We have proved that for the polynomial $f(x)=a_{1}x^{1}+\cdots +a_{k}x^{k}$ with integral coefficients, $f(x)\bmod N$ permutes $\mathbb{Z}_{N}$ if and only if $f(x)\bmod N$ permutes $S_{\mu}$ for all $\mu \mid N$, where $S_{\mu}=\{0< t <N: \gcd(N,t)=\mu\}$ and $S_{N}=S_{0}=\{0\}$. Based on it, we give a lower bound of the differential uniformities for such permutation polynomials, that is, $\delta (f)\geq \frac{N}{\#S_{a}}$, where $a$ is the biggest nontrivial divisor of $N$. Especially, $f(x)$ can not be APN permutations over the residue class ring \mathbb{Z}_{N}$. It is also proved that $f(x)\bmod N$ and $(f(x)+x)\bmod N$ can not permute $\mathbb{Z}_{N}$ at the same time when $N$ is even.

**Category / Keywords: **foundations / permutation polynomial, residue class ring, Almost Perfect Nonlinear (APN)

**Date: **received 3 May 2013

**Contact author: **yuyuyin at 163 com

**Available format(s): **PDF | BibTeX Citation

**Version: **20130503:083324 (All versions of this report)

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