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Soutenance de thèse de Mohamed SAEED TAHA 18/12/17

12/18/2017 @ 10:00 - 13:00

Soutenance lundi 18 décembre à 10h00 à l’Université de Rouen Normandie, campus
du Madrillet à Saint-Étienne-du-Rouvray au département d’informatique,
en salle des séminaires.
L’exposé sera donné en anglais, et sera suivi d’un pot auquel vous êtes
bien évidemment conviés.

Intitulé : Approche Algébrique sur L’Équivalence de Codes

Composition du jury :

– M. Daniel AUGOT DR / INRIA Saclay, Laboratoire d’Informatique de
l’Ecole Polytechnique (LIX) Rapporteur
– Mme Magali BARDET MCF / LITIS Université de Rouen Normandie Encadrante
de thèse
– Mme Delphine BOUCHER MCF/ IRMAR Université de Rennes I Examinatrice
– M. Philippe GABORIT PR / XLIM MATHIS Université de Limoges Examinateur
– M. Ayoub OTMANI PR / LITIS Université de Rouen Normandie Directeur de
thèse
– M. Nicolas SENDRIER DR /INRIA Rapporteur
– M . Jean-Pierre TILLICH DR / INRIA Examinateur

Résumé :
Code equivalence problem plays an important role in coding theory and
code-based cryptography.
That is due to its significance in classification of codes and also
construction and cryptanalysis of code based cryptosystems. It is also
related to the long standing problem of graph isomorphism, a well-known
problem in the world of complexity theory.
We introduce a new method for solving code equivalence problem. We
develop algebraic approaches to solve the problem in its permutation and
diagonal versions. We build algebraic system by establishing relations
between generator matrices and parity check matrices of the equivalent
codes. We end up with system of multivariables of linear and quadratic
equations which can be solved using algebraic tools such as Groebner
basis and related techniques.
By using Groebner basis techniques we can solve the code equivalence but
the computation becomes complex as the length of the code increases. We
introduced several improvements such as block linearization and
Frobenius action. Using these techniques we identify many cases where
permutation equivalence problem can be solved efficiently. Our method
for diagonal equivalence solves the problem efficiently in small fields,
namely F3 and F4 . The increase in the field size results in an increase
in the number of variables in our algebraic system which makes it
difficult to solve.
We introduce a new reduction from permutation code equivalence when the
hull is trivial to graph isomorphism. This shows that this subclass of
permutation equivalence is not harder than graph isomorphism. Using this
reduction we obtain an algebraic system for graph isomorphism with
interesting properties in terms of the rank of the linear part and the
number of variables. We solve the graph isomorphism problem efficiently
for random graphs and also for some regular graphs such as Petersen,
Cubical and Wagner Graphs.

Mots-clés : code equivalence, Groebner basis, Graph isomorphism,
Frobenius action, classification of codes, code-based cryptography.

Détails

Date :
12/18/2017
Heure :
10:00 - 13:00
Publié dans
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