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2026年学术讲座预告（No.7）湖北大学郑大彬教授做报告

时间：2026-05-11 编辑：lixy 点击：

报告题目：A Generalized $\chi_n$-Function

报告摘要：The mapping $\chi_n$ from $\F_{2}^{n}$ to itself defined by $y=\chi_n(x)$ with $y_i=x_i+x_{i+2}(1+x_{i+1})$, where the indices are computed modulo $n$, has been widely studied for its applications in lightweight cryptography. However, $\chi_n $ is bijective on $\F_2^n$ only when $n$ is odd, restricting its use to odd-dimensional vector spaces over $\F_2$. To address this limitation, we introduce and analyze the generalized mapping $\chi_{n, m}$ defined by $y=\chi_{n,m}(x)$ with $y_i=x_i+x_{i+m} (x_{i+m-1}+1)(x_{i+m-2}+1) \cdots (x_{i+1}+1)$, where $m$ is a fixed integer with $m\nmid n$. To investigate such mappings, we further generalize $\chi_{n,m}$ to $\theta_{m, k}$, where $\theta_{m, k}$ is given by $y_i=x_{i+mk} \prod_{\substack{j=1,m \nmid j}}^{mk-1} \left(x_{i+j}+1\right),{\rm for }\, i\in \{0,1,\ldots,n-1\}$. We prove that these mappings generate an abelian group isomorphic to the group of units in $\F_2[z]/(z^{\lfloor n/m\rfloor +1})$. This structural insight enables us to construct a broad class of permutations over $\F_2^n$ for any positive integer $n$, along with their inverses. We rigorously analyze algebraic properties of these mappings, including their iterations, fixed points, and cycle structures. Additionally, we provide a comprehensive database of the cryptographic properties for iterates of $\chi_{n,m}$ for small values of $n$ and $m$. Finally, we conduct a comparative security and implementation cost analysis among $\chi_{n,m}$, $\chi_n$, $\cchi_n$ (Yanis et al., EUROCRYPT2025) and their variants, and prove Conjecture~1 proposed by Yanis et al. as a by-product of our study. Our results lead to generalizations of $\chi_n$, providing alternatives to $\chi_n$ and $\rcchi_n$.

报告人简介：郑大彬，理学博士。现为湖北大学数学与统计学学院教授、博士生导师、院长，中国工业与应用数学学会编码密码及相关理论专业委员会委员、中国数学会计算机数学专业委员会委员、湖北省数学会副理事长。2006年6月于中科院数学与系统科学研究院获博士学位，2009年6月至2012年4月在中科院信息安全国家重点实验室从事博士后研究工作，2015年3月至2016年2月在美国特拉华大学访问、学习。先后主持国家自然科学基金项目4项、国家重点研发计划子课题1项以及省部级项目多项。在《IEEE Transactions on Information Theory》《Design, Codes and Cryptography》《Finite Fields and Their Applications》《Science China Mathematics》等国内外学术期刊和国际会议上发表论文60多篇。获湖北省自然科学二等奖1项（排名第一），第31届国际符号与代数计算(ISSAC2006)年会杰出论文奖。