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.

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 F 2 m . 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 F 2 m to be searched for. Hence, this enables us to generate all 4 × 4 involutory MDS matrices over F 2 3 and F 2 4 . 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 F 2 3 , F 2 4 and F 2 8 in terms of XOR count. In this context, we present new 4 × 4 involutory MDS matrices over F 2 3 , F 2 4 and F 2 8 , 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 2 k × 2 k involutory MDS matrices, where k > 1. For k = 1, Hadamard matrix form can be used to generate all involutory MDS matrices.