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<title><string language="fre"><![CDATA[3.5. Lee and Brickell Algorithm]]></string></title>
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<string language="fre"><![CDATA[In this fifth session,
we will study a variant of information set decoding
proposed by Lee and Brickell. So, the main idea
consists in relaxing the Prange algorithm to amortize the
cost of the Gaussian elimination. So, instead of error patterns with all
positions on the left, we will allow error patterns of
the form given in the slide. So, in the left part we
have w-p coordinate to 1 and on the right hand side
we allow a small number p of positions to have a value 1. So, at each iteration, we
will simply enumerate all the possible k to p values
for the right hand side. Note that the Prange
algorithm corresponds to weight p = 0. The computational syndrome
decoding and the solution we are going to propose to it
has an additional parameter p which is an integer between 0 and w. As before, we repeat the
following: we pick a permutation matrix P, we compute
the systematic form of that matrix using a
Gaussian elimination, and next, we enumerate this
set and we check every element in that set
until we find one element of weight w-p. If this
happens, then we have a solution to our problem, we return it. The cost of the iteration
in that case will increase because in addition to the
Gaussian elimination we now have an enumeration cost
which is equal to (k,p). Now, the complexity analysis. The probability to obtain
an error pattern of that form in a specific iteration is equal to P? as given in the slide. It follows that the
expected number of iteration N? is the inverse of the probability. And as I said before, the
iteration cost is the sum of those two terms. The
total cost of the Lee and Brickell algorithm is the
following: the product of N? * K, and in fact, it appears that we never gain more than a
polynomial factor compare with Prange algorithm, and I give here
the idea of how to prove this fact. And the last thing to do in
this formula is to minimize it over all possible
values of p and, except for very strange
parameters that's either w or W, the minimum value of that
formula is obtained for p=2.]]></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[algorithmes]]></string></keyword>
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<string language="fre"><![CDATA[3: Message Attacks (ISD)]]></string>
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