A Generalized $χ_n$-Function

2025-09-25

AI Summary

Existing χₙ mappings are bijective only over 𝔽₂ⁿ for odd n, limiting their applicability in even-dimensional lightweight cryptography. Method: We introduce the generalized mapping χₙ,ₘ and its further extension θₘ,ₖ, enabling invertible permutations over 𝔽₂ⁿ for arbitrary positive integers n. Through algebraic analysis, we establish that χₙ,ₘ forms an abelian group isomorphic to the unit group of a polynomial ring; we explicitly derive both the permutation and its inverse, and rigorously prove a conjecture recently posed at EUROCRYPT 2025. Results: We systematically characterize the iteration structure, fixed points, and cycle decomposition of χₙ,ₘ, and construct the first cryptographic property database for χₙ,ₘ iterations under small parameters. Our construction achieves superior security guarantees and reduced hardware implementation cost compared to state-of-the-art alternatives.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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 $m mid n$. To investigate such mappings, we further generalize $χ_{n,m}$ to $θ_{m, k}$, where $θ_{m, k}$ is given by $y_i=x_{i+mk} prod_{substack{j=1,,, m mid j}}^{mk-1} left(x_{i+j}+1 ight), ,,{ m for },, iin {0,1,ldots,n-1}$. We prove that these mappings generate an abelian group isomorphic to the group of units in $F_2[z]/(z^{lfloor n/m floor +1})$. 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 $χχ_n$.

Problem

• Extending χₙ mapping to work on even-dimensional vector spaces over 𝔽₂
• Generalizing χₙ to χₙ,ₘ mappings with parameter m not dividing n
• Constructing permutations over 𝔽₂ⁿ for any positive integer n

Innovation

• Generalized mapping χ_n,m for even-dimensional spaces
• Abelian group structure enabling permutation construction
• Comparative analysis of cryptographic properties and costs