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<title><string language="fre"><![CDATA[2.6. Reducing the Key Size]]></string></title>
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<string language="fre"><![CDATA[In the next three sessions,
I will explain how to reduce the key size of
code-based cryptosystem. Circulant matrices are the
central point in many attempts to reduce the key size of
code-based cryptosystems since they provide
efficient representation. A circulant matrix is
a square matrix, its rows are obtained by
cyclically shifting the first row. An alternative representation of an
n-tuple of elements is using polynomial. Thus, this matrix can be
described by a polynomial. And the i-th row of a
circulant matrix can be expressed by this formula. Circulant
matrices are closed under product and sum. Thus, this operation
preserves cyclicity. So, we have the
following proposition. Circulant matrices of size
r, with elements in Fq are equivalent to
polynomials in this quotient ring.
Block-Circulant matrices
are formed by concatenating circulant blocks of identical size. Quasi-cyclic codes have
been defined as a linear code that admits a block-circulant matrix. 
We will consider
quasi-cyclic codes that can be written in block-circulant systematic form. Quasi-cyclic subcodes of BCH codes
were proposed by Gaborit in 2005. Take notice that these codes
can be efficiently decoded. Thus, there are suitable
families for code-based cryptosystem. This table presents the
parameters suggested by the author. Note that the key size for
the same level of security drops considerably compared to
the original scheme with Goppa code.  The minimum size for
Goppa code is around 700 000  bits for 80 bits of
security while with this security level with quasi-cyclic
subcode of BCH code, we just need 12 000 bits.]]></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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