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<title><string language="fre"><![CDATA[4.8. Attack against Algebraic Geometry codes]]></string></title>
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<string language="fre"><![CDATA[In this session, we will present an
attack against Algebraic Geometry codes (AG codes). Algebraic Geometry codes
is determined by a triple. First of all, an
algebraic curve of genus g, then a n-tuple of rational points
and then a divisor which has disjoint support from the n-tuple P. Then, the Algebraic
Geometry code is obtained by evaluating at P all
functions that belong to the vector space associated to the divisor E. Some properties of these
codes are nearly optimal codes, that is, their designed minimum
distance is nearly the optimal one. Moreover, the dual of an
AG-code is again an AG-code. What about using Algebraic
Geometry codes in code-based cryptography? Janwa and
Moreno suggest to use Algebraic Geometry codes for the
McEliece cryptosystem. This is a suitable proposal
since these codes are nearly optimal and have
efficient decoding algorithms. If we talk about codes over
curves of genus zero then we are talking about
generalized Reed-Solomon codes, as we will see in the next slides.
So, for a curve of genus 0, this proposal is broken. If we talk about codes over
curves of genus 1 and 2, then this proposal is
broken by Faure and Minder. However, this attack has
several drawbacks which makes it impossible to extend to a
higher genera. But there is an attack for the general case. We will explain here this
general attack. First over generalized Reed-Solomon
codes and then we will give an idea on how it works
for the general case. Recall that the
generalized Reed-Solomon codes are Algebraic Geometry codes
over curves of genus 0. Indeed, if we consider the
projective line, this curve has genus 0 and its
points are of the form (x:y) Now, we will consider P the n-tuple of points formed by these
points and we take E to be K-1 times the point at the infinity. A basis of the
vector space associated to this divisor is the following one. And if we evaluate this
basis at the points P, we get a generator matrix of this
AG code, which is also a generator matrix of a
generalized Reed-Solomon code of dimension k associated to the
pair (a,1), the all-ones vector.]]></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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<string language="fre"><![CDATA[4: Key Attacks]]></string>
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