On the Construction Structures of 3 × 3 Involutory MDS Matrices Over F2m

Abstract

In this paper, we propose new construction structures, in other words, transposition-permutation path patterns for 3 × 3 involutory and MDS permutation-equivalent matrices over F23 and F24. We generate 3 × 3 involutory and MDS matrices over F23 and F24 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 × 3 involutory and MDS matrices from any 3 × 3 involutory and MDS representative matrix over F23 and F24. The 35 patterns are in common with these finite fields. By using these new transposition-permutation path patterns, new 3 × 3 involutory and MDS matrices can be generated especially for different finite fields such as F28 (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 × 8, 16 × 16, and 32 × 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. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.