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<title><string language="fre"><![CDATA[4.9. Goppa codes still resist]]></string></title>
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<string language="fre"><![CDATA[All the results that we 
have seen this week doesn't mean that code based
cryptography is broken. So in this session we will
see that Goppa code still resists to all these attacks. So recall that it is
assumed that Goppa codes are pseudorandom, that is
there exist no efficient distinguisher for Goppa
code. An efficient distinguisher was built for the case of high rate codes, where the rate
is very close to 1, but no generalization of this
distinguisher is known. The best known attacks are
based on the Support Splitting Algorithm and have
exponential runtime. In the third session of
this week, we have seen that Generalized Reed-Solomon codes
behave differently than random codes, with respect to the square
product that is the dimension of the square of a
Generalized Reed-Solomon code is very small compared to what it's
expected for a random code of the same length and dimension. Since an alternant code is
a subfield subcode of a Generalized Reed-Solomon
code, we might suspect that the star product of alternant codes also
behave differently from random codes. As we will see, this is true
but just for a very few cases. The following proposition
shows that the star product of two alternant codes is another
alternant code and the proof is very easy. We just need to recall
that alternant codes are subfield subcodes of
Generalized Reed Solomon code. So how works this proof?
Let c1 be a codeword of an alternant code and c2 be
another codeword of a different alternant code with the same support. Then, there exist two
polynomials of degree smaller than n-s and another polynomial
of degree smaller than n-r such that the evaluation
of these polynomials at the corresponding
elements give our codewords.]]></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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