1970-01-01

(2026-02) A Generalized -chi_n-Function

[[Cheng Lyu]]
[[Mu Yuan]]
[[Dabin Zheng]]
[[Siwei Sun]]
[[Shun Li]]

2026-02-10

Abstract

The mapping $\chi_n$ from $\mathbb{F}_{2}^{n}$ 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 $\mathbb{F}_2^n$ only when $n$ is odd, restricting its use to odd-dimensional vector spaces over $\mathbb{F}_2$ . 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 $\mathbb{F}_2[z]/(z^{\lfloor n/m\rfloor +1})$ . This structural insight enables us to construct a broad class of permutations over $\mathbb{F}_2^n$ 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 $\chi_{n,m}$ , $\chi_n$ , $\chi\chi_n$ and their variants, and prove Conjecture 1 proposed in [Belkheyar et al., 2025] as a by-product of our study. Our results lead to generalizations of $\chi_n$ , providing alternatives to $\chi_n$ and $\chi\chi_n$ .