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

Abstract

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 F2m and MDS matrices over F2mt ,where t ≥ 1 and m > 1, which preserves the software implementation properties in view of XOR operations and tablelookups of any given MDS matrix over F2m . Then we propose a novel method to obtain distinct functions relatedto these automorphisms and isomorphisms to be used in generating isomorphic MDS matrices (new MDS matrices inview 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 F28 (froman involutory 4 × 4 MDS matrix over F24 ) whose required number of XOR operations is the same as that of ANUBISand the subfield-Hadamard construction, and better than that of AES. The proposed method, due to its ground fieldstructure, is intended to be a complementary method for the current construction methods in the literature.