Suspension polymerization: Difference between revisions

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An '''MDS matrix (Maximum Distance Separable)''' is a [[matrix (mathematics)|matrix]] representing a function with certain [[diffusion (cryptography)|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]]
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of a [[linear transformation]] f(x)=Ax from K<sup>n</sup> to K<sup>m</sup> 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 [[Maximum distance separable code|MDS code]], i.e. a [[linear code]] that reaches the [[Singleton bound]].
 
Let <math>\tilde A = \left(\begin{array}{c}{\rm Id}_n\\  \hline{\rm A}\end{array}\right)</math> be the matrix obtained by joining the [[identity matrix]] Id<sub>n</sub> to A.
Then a necessary and sufficient condition for a matrix A to be MDS is that every possible n&times;n [[submatrix]] obtained by removing m rows from <math>\tilde A</math>
is [[non-singular matrix|non-singular]].
 
[[Reed-Solomon code]]s 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 primitive]]s 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 [[cryptanalysis|cryptanalyze]] functions that are not multipermutations.
 
MDS matrices are used for diffusion in such [[block cipher]]s as [[Advanced Encryption Standard|AES]], [[SHARK]], [[Square (cipher)|Square]], [[Twofish]], [[Anubis (cipher)|Anubis]], [[KHAZAD]], [[Manta (cipher)|Manta]], [[Hierocrypt]], and [[Camellia (cipher)|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
    | conference = 2nd International Workshop on [[Fast Software Encryption]] (FSE '94)
    | pages = 286&ndash;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
    | conference = 3rd International Workshop on Fast Software Encryption (FSE '96)
    | pages = 99&ndash;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 (cryptanalyst)|John Kelsey]], Doug Whiting, [[David A. Wagner|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 }}
 
{{crypto-stub}}
 
[[Category:Cryptography]]

Latest revision as of 20:28, 18 December 2014

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