RESUMO
This article presents a new hybrid method (combining search based methods and direct construction methods) to generate all 4×4 involutory maximum distance separable (MDS) matrices over F2m. The proposed method reduces the search space complexity at the level of n, where n represents the number of all 4×4 invertible matrices over F2m to be searched for. Hence, this enables us to generate all 4×4 involutory MDS matrices over F23 and F24. 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 F23, F24 and F28 in terms of XOR count. In this context, we present new 4×4 involutory MDS matrices over F23, F24 and F28, 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 2k×2k involutory MDS matrices, where k > 1. For k = 1, Hadamard matrix form can be used to generate all involutory MDS matrices.
