A Systematic Construction Approach for All 4× 4 Involutory MDS Matrices

Maximum distance separable (MDS) matrices play a crucial role not only in coding theory but also in the design of block ciphers and hash functions. Of particular interest are involutory MDS matrices, which facilitate the use of a single circuit for both encryption and decryption in hardware implementations. In this article, we present several characterizations of involutory MDS matrices of even order. Additionally, we introduce a new matrix form for obtaining all involutory MDS matrices of even order and compare it with other matrix forms available in the literature. We then propose a technique to systematically construct all 4 × 4 involutory MDS matrices over a finite field F2ᵐ. This method significantly reduces the search space by focusing on involutory MDS class representative matrices, leading to the generation of all such matrices within a substantially smaller set compared to considering all 4 × 4 involutory matrices. Specifically, our approach involves searching for these representative matrices within a set of cardinality (2ᵐ-1)⁵. Through this method, we provide an explicit enumeration of the total number of 4 × 4 involutory MDS matrices over F2ᵐ for m=3,4,…,8.