{"ID":2865847,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.20880","arxiv_id":"2509.20880","title":"A Generalized $χ_n$-Function","abstract":"The mapping $χ_n$ from $\\F_{2}^{n}$ to itself defined by $y=χ_n(x)$ 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, $χ_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 $χ_{n, m}$ defined by $y=χ_{n,m}(x)$ 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 $χ_{n,m}$ to $θ_{m, k}$, where $θ_{m, k}$ 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 $χ_{n,m}$ for small values of $n$ and $m$. Finally, we conduct a comparative security and implementation cost analysis among $χ_{n,m}$, $χ_n$, $χχ_n$ (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 $χ_n$, providing alternatives to $χ_n$ and $χχ_n$.","short_abstract":"The mapping $χ_n$ from $\\F_{2}^{n}$ to itself defined by $y=χ_n(x)$ 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, $χ_n $ is bijective on $\\F_2^n$ only when $n$ is odd, restricting its use to odd-dimensiona...","url_abs":"https://arxiv.org/abs/2509.20880","url_pdf":"https://arxiv.org/pdf/2509.20880v1","authors":"[\"Cheng Lyu\",\"Mu Yuan\",\"Dabin Zheng\",\"Siwei Sun\",\"Shun Li\"]","published":"2025-09-25T08:10:02Z","proceeding":"cs.CR","tasks":"[\"cs.CR\",\"cs.IT\"]","methods":"[]","has_code":false}