A Generalized $χ_n$-Function

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 $χχ_n$.