Title: A Generalized $χ_n$-Function Show Chinese title

Title: 一种广义的$χ_n$-函数

Abstract: 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 $\chi_{n,m}$, $\chi_n$, $\chi\chi_n$ (EUROCRYPT 2025 \cite{belkheyar2025chi}) and their variants, and prove Conjecture~1 proposed in~\cite{belkheyar2025chi} as a by-product of our study. Our results lead to generalizations of $\chi_n$, providing alternatives to $\chi_n$ and $\chi\chi_n$. Show Chinese abstract

Abstract: 从$\F_{2}^{n}$到自身的映射$\chi_n$由$y=\chi_n(x)$定义，其中$y_i=x_i+x_{i+2}(1+x_{i+1})$，其中索引是模$n$计算的，在轻量级密码学中的应用已被广泛研究。 然而，$\chi_n $仅在$n$为奇数时在$\F_2^n$上为双射，这将其应用限制在$\F_2$上的奇数维向量空间。 为解决这一限制，我们引入并分析由$y=\chi_{n,m}(x)$定义的广义映射$\chi_{n, m}$，其中$y_i=x_i+x_{i+m} (x_{i+m-1}+1)(x_{i+m-2}+1) \cdots (x_{i+1}+1)$，其中$m$是一个固定整数，满足$m\nmid n$。 为了研究这样的映射，我们进一步将$\chi_{n,m}$推广为$\theta_{m, k}$，其中$\theta_{m, k}$由$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\}$给出。 我们证明这些映射生成一个与$\F_2[z]/(z^{\lfloor n/m\rfloor +1})$中单位群同构的阿贝尔群。 这种结构上的洞察使我们能够构造出针对任意正整数$n$的$\F_2^n$上的一类广泛排列及其逆元。 我们严格分析了这些映射的代数性质，包括它们的迭代、不动点和循环结构。 此外，我们提供了针对 $\chi_{n,m}$ 的迭代在 $n$ 和 $m$ 的小值时的密码学性质的综合数据库。 最后，我们对$\chi_{n,m}$、$\chi_n$、$\chi\chi_n$(EUROCRYPT 2025\cite{belkheyar2025chi}) 及其变体进行了比较安全性和实现成本分析，并在我们的研究中作为副产品证明了~\cite{belkheyar2025chi}中提出的猜想~1。 我们的结果导致了$\chi_n$的推广，提供了$\chi_n$和$\chi\chi_n$的替代方法。