A Generalized $\ddot{I}{‡}_n$-Function

Abstract

The mapping $\ddot{I}{‡}_n$ from $\F_{2}^{n}$ to itself defined by $y=\ddot{I}{‡}_n(x)$ 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, $\ddot{I}{‡}_n$ 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 $\ddot{I}{‡}_{n,m}$ defined by $y=\ddot{I}{‡}_{n,m}(x)$ 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 $\ddot{I}{‡}_{n,m}$ to $\hat{I}{¸}_{m,k}$, where $\hat{I}{¸}_{m,k}$ is given by $y_i=x_{i+mk}{\prod }_{\begin{array}{c}j=1,m\nmid j\end{array}}^{mk-1}\left(x_{i+j}+1\right),\mathrm{for}i\in \{0,1,\dots ,n-1\}$. 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 $\ddot{I}{‡}_{n,m}$ for small values of $n$ and $m$. Finally, we conduct a comparative security and implementation cost analysis among $\ddot{I}{‡}_{n,m}$, $\ddot{I}{‡}_n$, $\ddot{I}‡\ddot{I}{‡}_n$ (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 $\ddot{I}{‡}_n$, providing alternatives to $\ddot{I}{‡}_n$ and $\ddot{I}‡\ddot{I}{‡}_n$.