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<title><string language="fre"><![CDATA[2.3. McEliece Assumptions]]></string></title>
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<string language="fre"><![CDATA[In this session, we will
talk about McEliece assumptions. The security of the
McEliece scheme is based on two assumptions as we have
already seen: the hardness of decoding a random linear
code and the problem of distinguishing a code with a
prescribed structure from a random one. In this sequence, we will study
in detail these two assumptions. The first assumption claims that
decoding a random linear code is difficult.  First, notice that the
general decoding problem is basically a re-writing of
the Syndrome Decoding problem. And both are equivalent
to the problem of finding codewords of minimal weight.
The Syndrome Decoding in the binary case is state as follows. Given a binary matrix, a
syndrome S and a non-negative integer W, the weight. The decision
problem faces the following question. There exists an error
pattern of weight at most w with syndrome S? while the computational
problem is to find such a vector. The decoding problem was
proved to be NP-complete in 1978 by Berlekamp, McEliece and
van Tilborg in this article. For the q-ary case,
see the article of Barg. Take notice that this proof
took place at the same time that the McEliece
cryptosystem was introduced. Thus, the worst case of
the computational problem is known to be difficult in general. Of course, depending on the
input, some instances can be solved in polynomial time as we
have already seen in the first week. Actually, the instance of
Syndrome Decoding involved in breaking code-based systems
are in particularly a subclass of Syndrome Decoding where the weight w is
bounded by half the minimum distance. This problem is not NP,
however it is conjectured to be NP-Hard. But even more in the
McEliece cryptosystem, the chosen code is not completely random. Even if the matrix is not
distinguishable from a random binary matrix of the same
size, the decoding problem uses parameters of those of a Goppa code. This means that the code has
length 2^m and the dimension is n mt, where t is
the correction capacity.]]></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[McEliece]]></string></keyword><keyword><string language="fre"><![CDATA[LDPC]]></string></keyword><keyword><string language="fre"><![CDATA[MDPC]]></string></keyword>
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