A generalization of the map $\chi$

Abstract

The mapping ${\chi }_n:{\mathbb{F}}_2^n\to {\mathbb{F}}_2^n$ defined by $y={\chi }_n(x)$ 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 $a_i\in {\mathbb{F}}_2$, $1\le i\le n$.

added a few remarks