In the literature, there are many results about permutation polynomials over finite fields. However, very few permutations of vector spaces are constructed although it has been shown that permutations of vector spaces have many applications in cryptography, especially in constructing permutations with low differential and boomerang uniformities. In this paper, motivated by the butterfly structure \cite{perrin2016cryptanalysis} and the work of Qu and Li \cite{qu2023}, we investigate rotatable permutations from $\gf_{2^m}^3$ to itself with $d$-homogenous functions. Based on the theory of equations of low degree, the resultant of polynomials, and some skills of exponential sums, we construct five infinite classes of $3$-homogeneous rotatable permutations from $\gf_{2^m}^3$ to itself, where $m$ is odd. Moreover, we demonstrate that the corresponding permutation polynomials of $\gf_{2^{3m}}$ of our newly constructed permutations of $\gf_{2^m}^3$ are QM-inequivalent to the known ones.

翻译：文献中已有许多关于有限域上置换多项式的结果。然而，尽管已经表明向量空间的置换在密码学中具有广泛应用（特别是在构造低差分和低回旋均匀度的置换方面），但构造的向量空间置换却非常少。本文受蝴蝶结构\cite{perrin2016cryptanalysis}以及Qu和Li的工作\cite{qu2023}的启发，研究了$\gf_{2^m}^3$上具有$d$-齐次函数的自可旋转置换。基于低次方程理论、多项式结式以及指数和技巧，我们构造了从$\gf_{2^m}^3$到自身的五类无限族$3$-齐次可旋转置换，其中$m$为奇数。此外，我们证明了新构造的$\gf_{2^m}^3$置换所对应的$\gf_{2^{3m}}$上的置换多项式与已知的置换多项式是QM不等价的。