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<title><string language="fre"><![CDATA[4.5. Error-Correcting Pairs]]></string></title>
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<description>
<string language="fre"><![CDATA[We present in this session a
general decoding method for linear codes. And we will see it in an example. Let C be a generalized
Reed-Solomon code of dimension k associated to the pair (c, d). Then, its dual is again a
generalized Reed-Solomon code with the same locator
and another column multiplier we will denote by d^ (d dual). Now, consider the codes A and B.  These codes have not
been chosen at random. First, notice that the star product
of these two codes is the dual of C. Suppose that these codes are
known. We will present here an efficient
decoding algorithm for C. So, let y be the received
word, that is, it is a sum of  a valid codeword and an
error vector, and suppose that there have been at most t errors. We define the following
kernel that is the set of elements of the code A which are
the solutions of this equation. We will see that the kernel
associated to the received vector is equivalent to the
kernel associated to the error vector. And the reason is very simple. First, notice that the star
product of the codes A and B, as we have already
said, is the dual of C. Moreover, by definition, the
inner product of the code with its dual is 0.  So, the codeword would
not affect this system. So, what do we have?  We have that the kernel
associated to the received vector is equivalent to the
kernel of the received vector. This is because the star
product of A and B is the dual code. So, now, we look for a
nontrivial element on this kernel, that is, a nontrivial
solution of this system, where e is the error vector. Let us define, first, the
error locator polynomial associated to the vector
c, that is, the root of this polynomial indicate
the error position. The evaluation of this
element belongs to the code A if, and only if, the
dimension of A is greater than t.]]></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[code correcteur]]></string></keyword><keyword><string language="fre"><![CDATA[algorithmes]]></string></keyword><keyword><string language="fre"><![CDATA[GRS code]]></string></keyword>
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REV:2021-07-06 18:02:55
FN:Irene MARQUEZ-CORBELLA
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<date><dateTime>2015-05-05</dateTime></date>
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<date><dateTime>2015-05-05</dateTime></date>
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<entity><![CDATA[BEGIN:VCARD
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FN:Matthieu FINIASZ
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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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<description>
<string language="fre"><![CDATA[4: Key Attacks]]></string>
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<string language="eng">DDC 22nd ed.</string>
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<string language="fre"><![CDATA[Analyse numérique]]></string>
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<string language="eng">DDC 22nd ed.</string>
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<string language="fre"><![CDATA[Théorie de l'information]]></string>
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<string language="eng">DDC 22nd ed.</string>
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