On the automorphisms and isomorphisms of MDS matrices and their efficient implementations

In this paper, we explicitly define the automorphisms of MDS matrices over the same binary extension field. By extending this idea, we present the isomorphisms between MDS matrices over F_2ᵐ and MDS matrices over F_2ᵐᵗ, where t ≥ 1 and $m>1$, which preserves the software implementation properties in view of XOR operations and table lookups of any given MDS matrix over F_2ᵐ. Then we propose a novel method to obtain distinct functions related to these automorphisms and isomorphisms to be used in generating isomorphic MDS matrices (new MDS matrices in view of implementation properties) using the existing ones. The comparison with the MDS matrices used in AES, ANUBIS, and subfield-Hadamard construction shows that we generate an involutory 4 × 4 MDS matrix over F_2⁸ (from an involutory 4 × 4 MDS matrix over F_2⁴) whose required number of XOR operations is the same as that of ANUBIS and the subfield-Hadamard construction, and better than that of AES. The proposed method, due to its ground field structure, is intended to be a complementary method for the current construction methods in the literature.