Constructing rotatable permutations of $\mathbb{F}_{2^m}^3$ with $3$-homogeneous functions
Yunwen Chi, Kangquan Li, Longjiang Qu
TL;DR
This work advances the construction of permutations on vector spaces by focusing on rotatable, $3$-homogeneous mappings on $\mathbb F_{2^m}^3$ for odd $m$. By exploiting rotation symmetry and employing a combination of resultant theory and exponential-sum analysis, the authors build five infinite families of such permutations with explicit $3$-homogeneous base polynomials $f$. They demonstrate that each family yields a permutation of $\mathbb F_{2^m}^3$ and that the corresponding permutation polynomials over $\mathbb F_{2^{3m}}$ are QM-inequivalent to known examples, due to higher-term complexity. The methods connect multivariate polynomial elimination, trace properties, and QM-equivalence concepts, and they enrich the set of vector-space permutations with potential cryptographic relevance, while leaving cryptographic properties such as differential/boomerang uniformities for future study.
Abstract
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.