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Title:
Permutation Polynomials and Their Differential Properties over Residue Class Rings
Authors: Yuyin Yu, 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.
ePrint: https://eprint.iacr.org/2013/251
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