<?xml version="1.0" encoding="UTF-8"?><lom xmlns="http://ltsc.ieee.org/xsd/LOM" xmlns:lomfr="http://www.lom-fr.fr/xsd/LOMFR" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://ltsc.ieee.org/xsd/LOM http://www.lom-fr.fr/xsd/lomfrv1.0/std/lomfr.xsd">
<general>
<identifier>
<catalog>Canal-U_Ocms</catalog>
<entry>32951</entry>
</identifier>
<title><string language="fre"><![CDATA[4.7. Attack against Reed-Muller codes]]></string></title>
<language>ENG</language>
<description>
<string language="fre"><![CDATA[In this session, we will
introduce an attack against binary Reed-Muller codes. Reed-Muller codes were
introduced by Muller in 1954 and, later, Reed provided the
first efficient decoding algorithm for these codes. Reed-Muller are just a
generalization of generalized Reed-Solomon codes. Generalized Reed-Solomon
codes are evaluation of univariate polynomials,
and Reed-Muller codes are evaluation of
multivariate polynomials. We will study binary Reed-Muller codes. The binary Reed-Muller
consists of the set of codewords obtained by evaluating all the
Boolean functions of degree r with m variables. Thus, the
block length of this code is 2^m. The dimension is a
number of polynomials of degree r with binary coefficient, and
the minimum distance is 2^m - r. Let us study two examples. The first example is a
Reed-Muller associated to the evaluation of all monomials
of degree 1 in three variables. The vectors associated
to these monomials are the following ones, which gives a
generator matrix of the code. This code has parameters:
length 8, dimension 4, and minimum distance 4, that
is, it detects two errors and correct just one error.
Take notice that the matrix of a Reed-Muller code with
degree bound 1 has a particular form. If we remove the first
row, then we have at  the i-th column just the number i-1
read as a binary number. And this property will
be the key of the attack. Now, we have another example. We have the binary
Reed-Muller code associated to the evaluation of monomials
of degree up to 2 in four variables. The vectors
associated to this monomial are the following ones, which give
a generator matrix of the code. This code has length 16,
dimension 11, and minimum distance 4. Let us see some
properties of Reed-Muller code. First of all, we have the
following decreasing sequence. The code with the same degree
bound as variables is the whole space. And moreover, the dual of a
Reed-Muller code is again a Reed-Muller code. And the proof is very easy.
Just note that the dimension of the sum of these two
codes is 2^n, that is, the whole space. Moreover, the star
product of these two codes is
 - this special code, which is
the code with all even weight vectors.]]></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>
<lomfr:documentType>
<lomfr:source>LOMFRv1.0</lomfr:source>
<lomfr:value>image en mouvement</lomfr:value>
</lomfr:documentType>
</general><lifeCycle>
<contribute>
<role>
<source>LOMv1.0</source>
<value>author</value>
</role>
<entity><![CDATA[BEGIN:VCARD
VERSION:3.0
CLASS:PUBLIC
REV:2021-07-06 18:02:55
FN:Irene MARQUEZ-CORBELLA
N:MARQUEZ-CORBELLA;Irene;;;
URL;TYPE=work:https://www.canal-u.tv/auteurs/marquez_corbella_irene
ROLE:author
TZ:+0200
END:VCARD
]]></entity>
<date><dateTime>2015-05-05</dateTime></date>
</contribute>
<contribute>
<role>
<source>LOMv1.0</source>
<value>author</value>
</role>
<entity><![CDATA[BEGIN:VCARD
VERSION:3.0
CLASS:PUBLIC
REV:2021-07-06 18:02:55
FN:Nicolas SENDRIER
N:SENDRIER;Nicolas;;;
URL;TYPE=work:https://www.canal-u.tv/auteurs/sendrier_nicolas
ROLE:author
TZ:+0200
END:VCARD
]]></entity>
<date><dateTime>2015-05-05</dateTime></date>
</contribute>
<contribute>
<role>
<source>LOMv1.0</source>
<value>author</value>
</role>
<entity><![CDATA[BEGIN:VCARD
VERSION:3.0
CLASS:PUBLIC
REV:2021-07-06 18:02:55
FN:Matthieu FINIASZ
N:FINIASZ;Matthieu;;;
URL;TYPE=work:https://www.canal-u.tv/auteurs/finiasz_matthieu
ROLE:author
TZ:+0200
END:VCARD
]]></entity>
<date><dateTime>2015-05-05</dateTime></date>
</contribute>
</lifeCycle>
<metaMetadata>
<metadataSchema>LOMv1.0</metadataSchema>
<metadataSchema>LOMFRv1.0</metadataSchema>
</metaMetadata>
<technical>
<format>video/mp4</format>
<location><![CDATA[https://www.canal-u.tv/video/inria/4_7_attack_against_reed_muller_codes.32951]]></location>
<location><![CDATA[https://streaming-canal-u.fmsh.fr/vod/media/canalu/videos/fuscia/4.6.attack.against.grs.codes.copie._32951/c015im.w4.s7.mov]]></location>
<size>159084003</size>
<duration><duration>PT0H5M48S</duration></duration>
</technical>
<educational>
<learningResourceType>
<source>LOMv1.0</source>
<value>lecture</value>
</learningResourceType>
<context>
<source>LOMv1.0</source>
<value>master</value>
</context>
<context>
<source>LOMv1.0</source>
<value>doctorat</value>
</context>
</educational>
<rights>
<cost>
<source>LOMv1.0</source>
<value>no</value>
</cost>
<copyrightAndOtherRestrictions>
<source>LOMv1.0</source>
<value>no</value>
</copyrightAndOtherRestrictions>
<description>
<string language="fre"><![CDATA[Droits réservés à l'éditeur et aux auteurs. 
Ces ressources de cours sont, sauf mention contraire, diffusées sous Licence Creative Commons. L’utilisateur doit mentionner le nom de l’auteur, il peut exploiter l’œuvre sauf dans un contexte commercial et il ne peut apporter de modifications à l’œuvre originale.]]></string>
</description>
</rights>
<relation>
<kind>
<source>LOMv1.0</source>
<value>ispartof</value>
</kind>
<resource>
<identifier>
<catalog>URI</catalog>
<entry>https://www.canal-u.tv/producteurs/inria/cours_en_ligne/code_based_cryptography/4_key_attacks</entry>
</identifier>
<description>
<string language="fre"><![CDATA[4: Key Attacks]]></string>
</description>
</resource>
</relation>
<classification>
<purpose>
<source>LOMv1.0</source>
<value>discipline</value>
</purpose>
<taxonPath>
<source>
<string language="fre"><![CDATA[Universités Numériques Thématiques 2009 http://www.universites-numeriques.fr]]></string>
</source>
<taxon>
<id/>
<entry>
<string language="fre"/>
</entry>
</taxon>
</taxonPath>
</classification>
<classification>
<purpose>
<source>LOMv1.0</source>
<value>discipline</value>
</purpose>
<taxonPath>
<source>
<string language="fre">CDD 22e éd.</string>
<string language="eng">DDC 22nd ed.</string>
</source>
<taxon>
<id>518</id>
<entry>
<string language="fre"><![CDATA[Analyse numérique]]></string>
</entry>
</taxon>
</taxonPath>
<taxonPath>
<source>
<string language="fre">CDD 22e éd.</string>
<string language="eng">DDC 22nd ed.</string>
</source>
<taxon>
<id>003.54</id>
<entry>
<string language="fre"><![CDATA[Théorie de l'information]]></string>
</entry>
</taxon>
</taxonPath>
<taxonPath>
<source>
<string language="fre">CDD 22e éd.</string>
<string language="eng">DDC 22nd ed.</string>
</source>
<taxon>
<id>005.7</id>
<entry>
<string language="fre"><![CDATA[données dans les systèmes informatiques]]></string>
</entry>
</taxon>
</taxonPath>
<taxonPath>
<source>
<string language="fre">CDD 22e éd.</string>
<string language="eng">DDC 22nd ed.</string>
</source>
<taxon>
<id>652.8</id>
<entry>
<string language="fre"><![CDATA[cryptographie]]></string>
</entry>
</taxon>
</taxonPath>
<taxonPath>
<source>
<string language="fre">CDD 22e éd.</string>
<string language="eng">DDC 22nd ed.</string>
</source>
<taxon>
<id>510</id>
<entry>
<string language="fre"><![CDATA[Mathématiques]]></string>
</entry>
</taxon>
</taxonPath>
</classification> </lom>