A Class of Large-Scale MDS Constructions Based on Known MDS Matrices

Symmetric ciphers are widely used due to their consistent encryption and decryption processes and high efficiency. The principles of confusion and diffusion are the two fundamental design principles of symmetric ciphers. The use of linear matrices as diffusion layers is a mainstream approach in symmetric cipher design. For instance, AES employs Maximum Distance Separable (MDS) codes, a type of linear error-correcting code, in order to construct the optimal diffusion layers. Lightweight MDS matrices with low Hamming weights and minimal implementation costs are preferred by designers. Examples include the cyclic matrice like(2,3,1,1) and another Hadamard matrice like (2,5,3,1). Traditional MDS constructions over the finite field rely on the selection of irreducible polynomials, which limits their scalability. In this paper, we have investigated the large-scale construction of MDS matrices based on existing ones. Through introducing a single-cycle permutation parameter, we have redefined the xtime(α) operation on the finite field, which is decoupling multiplication from cyclic shifts and XOR operations. This enables broader MDS constructions, expanding the existing MDS quantity by a factor of n!. Specific instances of MDS constructions for two class of matrices are provided.