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Concept# MDS matrix

Summary

An MDS matrix (maximum distance separable) is a matrix representing a function with certain diffusion properties that have useful applications in cryptography. Technically, an m \times n matrix A over a finite field K is an MDS matrix if it is the transformation matrix of a linear transformation f(x) = Ax from K^n to K^m such that no two different (m + n)-tuples of the form (x, f(x)) coincide in n or more components.
Equivalently, the set of all (m + n)-tuples (x, f(x)) is an MDS code, i.e., a linear code that reaches the Singleton bound.
Let \tilde A = \begin{pmatrix} \mathrm{I}_n \ \hline \mathrm{A} \end{pmatrix} be the matrix obtained by joining the identity matrix \mathrm{I}_n to A. Then a necessary and sufficient condition for a matrix A to be MDS is that every possible n \times n

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