A new hybrid method combining search and direct based construction ideas to generate all 4 × 4 involutory maximum distance separable (MDS) matrices over binary field extensions.

Abstract

This article presents a new hybrid method (combining search based methods and direct construction methods) to generate all involutory maximum distance separable (MDS) matrices over . The proposed method reduces the search space complexity at the level of , where n represents the number of all invertible matrices over to be searched for. Hence, this enables us to generate all involutory MDS matrices over and . After applying global optimization technique that supports higher Exclusive-OR (XOR) gates (e.g., XOR3, XOR4) to the generated matrices, to the best of our knowledge, we generate the lightest involutory/non-involutory MDS matrices known over , and in terms of XOR count. In this context, we present new involutory MDS matrices over , and , which can be implemented by 13 XOR operations with depth 5, 25 XOR operations with depth 5 and 42 XOR operations with depth 4, respectively. Finally, we denote a new property of Hadamard matrix, i.e., (involutory and MDS) Hadamard matrix form is, in fact, a representative matrix form that can be used to generate a small subset of all involutory MDS matrices, where k > 1. For k = 1, Hadamard matrix form can be used to generate all involutory MDS matrices.