MDS matrix

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×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 Kⁿ 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 be the matrix obtained by joining the identity matrix Id_n to A.Then a necessary and sufficient condition for a matrix A to be MDS is that every possible n×n submatrix obtained by removing m rows from $ilde\; A$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": changing t of the inputs changes at least m-t+1 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, Manta, Hierocrypt, and Camellia, and in the stream cipher MUGI and the cryptographic hash function WHIRLPOOL.

References

* cite conference
author = Serge Vaudenay
title = On the Need for Multipermutations: Cryptanalysis of MD4 and SAFER
booktitle = 2nd International Workshop on Fast Software Encryption (FSE '94)
pages = pp.286–297
publisher = Springer-Verlag
date = November 16 1994
location = Leuven
url = http://citeseer.ist.psu.edu/vaudenay94need.html
format = PDF/PostScript
accessdate = 2007-03-05
* cite conference
author = Vincent Rijmen, Joan Daemen, Bart Preneel, Anton Bosselaers, Erik De Win
title = The Cipher SHARK
booktitle = 3rd International Workshop on Fast Software Encryption (FSE '96)
pages = pp.99–111
publisher = Springer-Verlag
date = February 1996
location = Cambridge
url = http://citeseer.ist.psu.edu/rijmen96cipher.html
format = PDF/PostScript
accessdate = 2007-03-06
* cite paper
author = Bruce Schneier, John Kelsey, Doug Whiting, David Wagner, Chris Hall, Niels Ferguson
title = The Twofish Encryption Algorithm
date = June 15 1998
url = http://www.schneier.com/paper-twofish-paper.html
format = PDF/PostScript
accessdate = 2007-03-04