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<title><string language="fre"><![CDATA[2.8. Reducing the Key Size - MDPC codes]]></string></title>
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<string language="fre"><![CDATA[This is the last session where we
will talk about reducing the key size. Here we will
introduce the MDPC codes.
In 2012, the MDPC codes were
proposed for the McEliece schemes. An MDPC code is a
code that admits a binary
moderate density-parity check matrix. Typically, the Hamming
weight of each row is of the order the square of the length. In this sequence, I will
describe this scheme of quasi-cyclic MDPC McEliece for
a binary code of rate one half. So, we use circulant
matrices of blocks of size p to define the codes. The length
will be 2p and the dimension p. Other parameters are the
weight of the parity check equations and the number
of correctable errors. So, let us explain the McEliece
schemes using quasi-cyclic MDPC code. First of all, we pick
randomly two vectors of weight p,
such that the concatenated
vector has a weight smaller than w. We will repeat until the
corresponding polynomial h0 is invertible. In particular,
we ask the weight to be odd. Then, the secret key and the public
key will be the corresponding matrices. To encrypt a message, we
apply the following function, that is, we encode the
message and we add random errors of weight smaller than t. But we will describe
them in terms of polynomial. To decrypt, we use an
MDPC-like iterative decoding algorithm as the
Gallager's Bit-Flipping algorithm, already explained in
the previous session. The quasi-cyclic MDPC proposal
is secure under two assumptions. First of all, the problem of
distinguishing a public key from a random quasi-cyclic
matrix or equivalently the problem of finding
codewords of weight w in the dual of an MDCP code; and the
hardness of decoding random quasi-cyclic codes. The
security reduction can be translated in terms of
polynomials as follows.]]></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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<date><dateTime>2015-05-05</dateTime></date>
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<string language="fre"><![CDATA[2: McEliece Cryptosystem]]></string>
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