An MDS matrix (maximum distance separable) is a matrix representing a function with certain diffusion properties that have useful applications in cryptography over a finite field is an MDS matrix if it is the transformation matrix of a linear transformation is an MDS code, i.e., a linear code that reaches the Singleton bound.

be the matrix obtained by joining the identity matrix submatrix is non-singular.

Reed–Solomon codes have the MDS property and are frequently used to obtain the MDS matrices used in cryptographic algorithms.

Serge Vaudenay suggested using MDS matrices in cryptographic primitives to produce what he called multipermutations, not-necessarily linear functions with this same property. These functions have what he called perfect diffusion of the outputs. He showed how to exploit imperfect diffusion to cryptanalyze functions that are not multipermutations.

MDS matrices are used for diffusion in such block ciphers as AES, SHARK, Square, Twofish, Anubis, KHAZAD, Manta, Hierocrypt, Kalyna, Camellia and HADESMiMC, and in the stream cipher MUGI and the cryptographic hash function Whirlpool, Poseidon.

References