PreprintCC BY 4.0arXiv
A generalization of the map χ
Abstract
The mapping $ χ_n:\mathbb{F}_2^n \to \mathbb{F}_2^n$ defined by y=χn(x) with yi = xi + xi+1xi+2 + xi+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 yi=xi+u+xi+v(xi+w+ai), where 0≤ u, v, w<n and ai∈ 𝔽2, 1≤ i≤ n.
BibTeX
@misc{feng2026generalization,
title = {{A generalization of the map $χ$}},
author = {Xiutao Feng and Qiang Wang and Jingyi Yu and Anpeng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19548},
eprint = {2609.19548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19548},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A generalization of the map χ
AU - Xiutao Feng
AU - Qiang Wang
AU - Jingyi Yu
AU - Anpeng Zhang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.19548
UR - https://arxiv.org/abs/2609.19548
AB - 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+1x_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≤ u, v, w<n and a_i∈ 𝔽_2, 1≤ i≤ n.
ER -