On the Construction Structures of $3 \times 3$ Involutory MDS Matrices over $\mathbb {F}_{2^{m}}$

Abstract

In this paper, we propose new construction structures, in other words, transposition-permutation path patterns for $3 \times 3$ involutory and MDS permutation-equivalent matrices over $\mathbb {F}_{2^{3}}$ and $\mathbb {F}_{2^{4}}$. We generate $3 \times 3$ involutory and MDS matrices over $\mathbb {F}_{2^{3}}$ and $\mathbb {F}_{2^{4}}$ by using the matrix form given in [1], and then all these matrices are analyzed by finding all their permutation-equivalent matrices. After that, we extract whether there are any special permutation patterns, especially for this size of the matrix. As a result, we find new 28,088 different transposition-permutation path patterns to directly construct $3 \times 3$ involutory and MDS matrices from any $3 \times 3$ involutory and MDS representative matrix over $\mathbb {F}_{2^{3}}$ and $\mathbb {F}_{2^{4}}$. The 35 patterns are in common with these finite fields. By using these new transposition-permutation path patterns, new $3 \times 3$ involutory and MDS matrices can be generated especially for different finite fields such as $\mathbb {F}_{2^{8}}$ (is still an open problem because of the large search space). Additionally, the idea of finding the transposition-permutation path patterns can be applicable to larger dimensions such as $8 \times 8$, $16 \times 16$, and $32 \times 32$. To the best of our knowledge, the idea given in this paper to find the common and unique transposition-permutation path patterns over different finite fields is the first work in the literature.