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<title><string language="fre"><![CDATA[4.6. Attack against GRS codes]]></string></title>
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<string language="fre"><![CDATA[In this session we will
discuss the proposal of using generalized Reed-Solomon codes
for the McEliece cryptosystem. As we have already said,
generalized Reed-Solomon codes were proposed in
1986 by Niederreiter. Recall that these codes are
MDS, that is, they attain the maximum error correcting
capacity which is interpreted as shorter keys for the
same level of security. Moreover, these codes have
efficient decoding algorithms so they are suitable
candidates for code-based cryptography. But this proposal is
subject to a polynomial attack by Sidelnikov-Shestakov. Take notice that if we know
a generalized Reed-Solomon code associated to the pair
(a,b) of dimension k and dimension k - 1, that is
we know two codes, then by solving the following
system, we can obtain the generalized Reed-Solomon
code of dimension k - 2. And the proof is very easy. Just take notice that the
star product of the known codes provide the square code of
the generalized Reed-Solomon code of dimension k - 1. 
In other words, a polynomial of degree at most k-1
times a polynomial of degree at most k, yields to a
polynomial of degree at most 2k - 2. And this property can be
used iteratively to build the following decreasing sequence. Note that the generalized
Reed-Solomon code of dimension 1 is the set of
multiples of the vector b. Thus we can retrieve
the vector b and then by solving a linear
system, we can obtain also a. However, from the McEliece
cryptosystem, just a generator matrix of the generalized
Reed-Solomon code of dimension k is known. And we will explain in
this slide how to compute a generator matrix of the
same generalized Reed-Solomon code by dropping by
one the dimension. Recall that the shorten
of a generalized Reed-Solomon code is again a
generalized Reed-Solomon code. Moreover, the code locator
of the shortened code is again the same code locator but
restricted to some coordinates. In particular shortening at
the first position we get a new generalized Reed-Solomon code, associated to the same code locator without
the first coordinate, and it is easy to get a generator
matrix of such shortened code: we just apply Gaussian
Elimination to our matrix.]]></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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<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[4: Key Attacks]]></string>
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