Atelier Expérimental sous l'égide de la Fondation Européenne de la Science
Ecole Européenne G.A.T.I.
An European Science Fondation - Exploratory Workshop :
European School A.G.I.T.
Géométrie Algébrique et Théorie de l'Information
Algebraic Geometry and Information Theory
se tiendra au C.I.R.M. du 12 au 16 mai 2003
will take place at the C.I.R.M. may 12th - 16th 2003
Organizers : Yves Aubry, Gilles Lachaud, Mikhail Tsfasman
The school will be composed by the following four courses:
1. Mathematical Background of Public Key Cryptography (Gerhard Frey);
2. On Curves and Function Fields over finite fields (Arnaldo Garcia);
3. Sphere packings in Euclidean and Hamming spaces (Gregory Kabatianski*);
4. Elliptic curves over finite fields and algorithms (René Schoof).
*This course was originally planned for Neil Sloane however he expressed his inability to come because of the Orange Level security alert in the US !
Lecturers:
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Gerhard Frey Institut für Experimentelle Mathematik Universitäat GH Essen, Ellernstr. 29 45326 Essen, Germany frey@exp-math.uni-essen.de
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Arnaldo Garcia
Institute of Pure and Applied Mathematics (IMPA) Brazil garcia@impa.br
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René Schoof Dipartimento di Matematica Univ. di Roma "Tor Vergata" I-00185 Roma, Italia schoof@mat.uniroma2.it
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Gregory Kabatianski I.P.P.I. Institut pour les Problèmes de la Transmission des Informations Moscou, Russie kaba@iitp.ru
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Program:
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PROGRAM
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9h30-10h30
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.
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11h00-12h00
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.
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14h30-15h30
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.
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16h00-17h30
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Monday
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Schoof
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.
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Frey
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Schoof
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.
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Garcia
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Tuesday
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Schoof
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.
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Frey
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.
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Schoof
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Garcia
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Wednesday
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Schoof
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.
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Garcia
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.
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Excursion - Dinner
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Thursday
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Kabatianski
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.
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Frey
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.
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Kabatianski
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.
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Garcia
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Friday
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Kabatianski
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.
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Frey
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.
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14h00-15h00 : Kabatianski |
.
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15h15-16h30 : Garcia |
Abstract of the courses:
1. Mathematical Background of Public Key Cryptography (Gerhard Frey): (slides in pdf)
One of the most efficient tools in Public Key Cryptography are discrete logarithms. The mathematical task is to construct groups of large prime order in which it is easy to add but very difficult to compute the discrete logarithm.
We shall explain how one can use ideal class groups of rings of integers in global fields for this purpose. This will lead to the arithmetic of hyperelliptic curves and their Jacobian varieties over finite fields for which we shall establish explicit addition formulas.
The rich structure of these objects and especially the Galois action will imply both possibilities of attacks and of constructions relying on point counting by the use of l-adic and p-adic representations on spaces of differentials.
The contents would be:
1.) Public Key Cryptography, Discrete Logarithms
2.) Computational Aspects of Picard Groups
3.) Duality of Abelian Varieties and Discrete Logarithms
4.) Point Counting by p-adic and l-adic Cohomology
2. On Curves and Function Fields over finite fields (Arnaldo Garcia):
Lect.1 Rational Points(Weil's Theorem). Description:Facts about the Zeta-Function,Serre's improvement and explicit formulas.Maximal Curves and Ihara's genus bound.Some results on the classification of maximal curves.
Lect.2 Constructions of good curves and Stohr-Voloch Theory. Description:methods by van der Geer-van der Vlugt and Garcia-Garzon-Quoos(the genus of Kummer covers of the projective line).The theory of Stohr-Voloch in the particular case of nonsingular plane curves.Examples.
Lect.3 Asymptotic Behaviour. Description:the Drinfeld-Vladut bound.Towers of Function Fields over finite fields and their limit.Tame Towers:Ramification set and Completely splitting set.Examples.
Lect.4 Towers of Artin-Schreier type.
Lect.5 Towers of Artin-Schreier type.
In the last two lectures I would try to give some recent results on this type of towers,showing in particular that one has to perform pole-order reductions infinitely many times if the tower is "good", and also to present a tower of this type attaining the Drinfeld-Vladut bound(Inventiones Math. 121(1995),211-222).
3. Sphere packings in Euclidean and Hamming spaces (Gregory Kabatianski):
1. Sphere packings in general metric spaces, Minkovski-Hlawka-Varshamov-Gilbert bound.
2. Spherical codes and error-correcting codes - a unified approach to upper bounds. Examples of optimal or near optimal packings.
3. Constructions of lattice packing of spheres via error-correcting codes.
4. Elliptic curves over finite fields and algorithms (René Schoof):
Assuming some basic properties of algebraic curves, the theory of elliptic curves over finite fields will be discussed. This includes in particular, the distribution of elliptic curves with respect to their number of points and a description of the various endomorphism rings that may occur. Several algorithms involving elliptic curves over finite fields will be studied.
Remerciement : Cette Ecole Européenne voit le jour grâce à l'European Science Fondation (http://www.esf.org).
Acknowledgment : The School is organized by the generous support from the European Science Fondation (http://www.esf.org).
Excursion: We have planned for an excursion tour to the cave of ``Marie-Madeleine de la Sainte Beaume'' in the afternoon of wednesday. All the participants are welcome to join.
More details: the bus will take us from Luminy at 13h45 to a place called Plan d'Aups. From there, we will walk down until the cave of ``Marie-Madeleine de la Sainte Beaume''. Participants are urged to come with walking shoes. In the evening, the dinner will be at the ``Lou Pèbre d'Aï'' restaurant. We hope to get back to Luminy by 21h30.
Kindly inform us latest by monday afternoon if you are not interested to join.
Some pictures of the excursion : photos
List of participants: here
Frais de séjour :
Les frais de séjour des participants pourront éventuellement être pris en charge par l'organisation de l'Ecole, ceci dans la limite des fonds disponibles.
Hotel expenses:
The hotel expenses of the participants will be taken care by the organization of the School depending on the availability of the funds.
AGCT-9 conference:
There will be an international conference the week after the School G.A.T.I. called Arithmetic, Geometry and Coding Theory: click here.
Registration:
Renseignements pratiques :
Information: