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arXiv · 2609.19548

A generalization of the map $χ$

Xiutao Feng·Qiang Wang·Jingyi Yu·Anpeng Zhang
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Abstract

The mapping $ χ_n:\mathbb{F}_2^n \to \mathbb{F}_2^n$ defined by $y=χ_n(x)$ with $y_i = x_i + x_{i+1}x_{i+2} + x_{i+2}$, where the indices are computed modulo $n$, has been widely studied for its application in lightweight cryptography. In this paper, we generalize this mapping and completely characterize all these shift-invariant permutations of the form $y_i=x_{i+u}+x_{i+v}(x_{i+w}+a_i)$, where $0\le u, v, w<n$ and $a_i\in \mathbb{F}_2$, $1\le i\le n$.

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Xiutao Feng, Qiang Wang, Jingyi Yu, Anpeng Zhang. 2026-09-21. A generalization of the map $χ$. https://arxiv.org/abs/2609.19548

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