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<title><string language="fre"><![CDATA[3.3. Information Set Decoding: the Power of Linear Algebra]]></string></title>
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<string language="fre"><![CDATA[In this third session, we
will present the most important concept of the week:
Information Set Decoding. The problem of decoding is not
only a combinatorial problem. Because we are dealing with linear
code, we may also use Linear Algebra. In particular, we are able
to transform the Computational Syndrome Decoding
problem by multiplying the matrix by a
permutation P on the right and a nonsingular matrix U on the left. This will transform the
problem of syndrome decoding into an equivalent one. It is very easy to prove that
the following equivalence holds. In fact, this implies that
the two computational syndrome decoding that are
presented here are equivalent.
If I solve one of
them, I solve the other. In particular, I can
choose H'in the following form which is called
systematic form or standard form. If the first n-k columns of matrix H*P are
independent, then I am able to compute such a form. If this
is the case, the last k column of the matrix
forms an information set. If I am lucky, the
permutation P will send the w error position in
the left part of the position. If this happens,
then the modified syndrome s'will have a weight equal
to the weight of the error. And I can check that very easily. I can transform
that into an algorithm. So, I want to solve the
computational syndrome decoding presented here. I will repeatedly pick
a permutation P, compute the following systematic
form using Gaussian elimination, and check whether the
transformed syndrome s*(U transpose) is a weight w. When this hapens, I can
return the following vector which is valid
solution to the original syndrome decoding problem. Here, each iteration of this
algorithm will cost n*(n-k) column operations that is the
cost of the Gaussian elimination. And I have to repeat that
a certain number of times, until one of the
solutions to my problem has all its non-zero coordinates "all left". In the next session, I will present Complexity Analysis of that
algorithm in particular as long as tools to make the
complexity analysis of the variance of information
set decoding I will present throughout this course.]]></string></description>
<keyword><string language="fre"><![CDATA[algèbre linéaire]]></string></keyword><keyword><string language="fre"><![CDATA[chiffrement à clé publique]]></string></keyword><keyword><string language="fre"><![CDATA[cryptage des données]]></string></keyword><keyword><string language="fre"><![CDATA[cryptographie]]></string></keyword><keyword><string language="fre"><![CDATA[algorithmes]]></string></keyword>
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<date><dateTime>2015-05-05</dateTime></date>
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<string language="fre"><![CDATA[Droits réservés à l'éditeur et aux auteurs. 
Ces ressources de cours sont, sauf mention contraire, diffusées sous Licence Creative Commons. L’utilisateur doit mentionner le nom de l’auteur, il peut exploiter l’œuvre sauf dans un contexte commercial et il ne peut apporter de modifications à l’œuvre originale.]]></string>
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<string language="fre"><![CDATA[3: Message Attacks (ISD)]]></string>
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