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Théorie Algébrique des nombres et applications notamment à la cryptographie

Ecole CIMPA-ICTP, Abidjan 2017

Dates : du 10 avril au samedi 22 avril 2017

Lieu : Université Félix Houphouët Boigny





Les cours et conférences de l'école


Cours 1 : Application of the elementary Number Theory to cryptography (Alain TOGBE, Westville, USA)
This course will use the basics of the elementary number theory to introduce cryptography. So the theory of congruences will be gently used to introduce cryptography. From the Caesar cipher, we will present the public-key cryptography. We will also discuss the knapsack cryptosystem (which is based on the difficult classic problem in combinatorics known as the knapsack problem) and the discrete logarithm problem.

Cours 2 :Introduction to topics of algebraic number theory related to cryptography (Michel WALDSCHMIDT, Paris 6, France)
This course will introduce some of the main basic tools from number theory which are essential for the modern cryptographic systems. Starting with the arithmetic of rational integers, divisibility and congruences will be explained in connection with algebra (group theory, especially cyclic groups; rings, ideals, quotients; fields). The finite fields with a number of elements which is a prime number occur naturally in this context; however they do not suffice for advanced purposes, and the general theory of finite fields will be developed. Such a theory has a lot of deep applications, some of which will be outlined.

Cours 3 : Décomposition cyclotomique sur les corps finis (François Emmanuel TANOE, Abidjan, Côte d’Ivoire)
Ce cours a pour objectif de donner les outils nécessaires à l’étude des codes correcteurs d’erreurs, notamment à celle des codes de HAMMING. Ce cours utilise de l’arithmétique de base (Fonctions Multiplicatives, notamment la forme multiplicative des formules d’inversion de Möbius) ;de l’arithmétique modulaire, ainsi que de la Théorie de Galois. La cyclicité des groupes de Galois des extensions de corps finis sera essentielle pour étudier la décomposition en facteurs irréductibles des polynômes sur un corps finis. Des exemples de construction de tels corps finis sont donnés pour des petites valeurs de leurs cardinaux. La décomposition des polynômes cyclotomiques sur les corps finis sont détaillés et des exemples numériques nombreux sont donnés ainsi que les principaux Théorèmes fondamentaux.

Cours 4 : Lattices and Cryptography (Adriana SALERNO, Lewiston, USA)
- Review on vector spaces and lattices.
- The shortest vector problem and the closest vector problem.
- Babai's algorithm for finding "good" bases.
- Cryptosystems based on hard lattice problems: The GGH Public Key Cryptosystem, The NTRU Public Key Cryptosystem.

Possible additional topics:
- Lattice-based digital signature schemes.
- Lattice reduction algorithms. 
- Fully homomorphic encryption. 

Cours 5 : Elementary Approach to elliptic curves (Francesco PAPPALARDI, Roma, Italy)
Examples of elliptic curves, drawing elliptic curves, the set of rational points of an elliptic curve, intersection between a line and an elliptic curve, the point at infinity of an elliptic curve, singular points, the group law, Weierstrass equations and their classification, elliptic curves over finite fields and their properties, the Hasse bound, the structure of the group of points over finite fields.

Cours 6 : Introduction to the theory of modular forms and applications (Lassina DEMBELE, Warwick, UK)
Modular forms play a central role in modern number theory. In this series of lectures, we will give a gentle introduction to this beautiful theory. The only prerequisite for this course will be a basic knowledge of the theory of complex analysis in one variable. If time permit, we will discuss some applications to number theory and elliptic curves

TD: An introduction to PARI/GP (Christophe DELAUNAY, Besançon, France)
The main goal of this course is to give an introduction to the PARI/GP software which is number theoretic computational software. We will illustrate several notions given in the courses of this school (modular arithmetic, cyclotomic fields, elliptic curves, L-functions, ...) with an emphasize on the explicit and experimental point of views. We will insist on the cryptographic aspects and, in particular, we will explain how to build explicitly some classical crypto-systems.

Conférence 1 : Tableaux de Young et Cryptographie (Eric Dago AKEKE, Abidjan, Côte d’Ivoire)
The course will be mainly focused on Young tableaux and Cryptography. It will contain the necessary basic tools specific to Young tableaux. Basic concepts of cryptography will be revisited. Young's tables will be used, thereafter, to polygraph message encryption tools, with Key transfer via an asymmetric encryption (RSA).

Conférence 2 : Cryptographie et sécurité du signal et/ou de l’information (Prosper Kouadio KIMOU, Yamoussoukro, Côte d’Ivoire )
In this paper we first show how a signal is processed to be transmitted without error and without compromise. We then described how mathematical tools such as number theory, modular arithmetic and probability theory are used in cryptographic protocols or utilized. Furthermore, we explain and discuss with illustrations to support the implementation of cryptographic mechanisms and protocols to ensure the basic objectives of information security namely the availability, integrity and confidentiality. Finally, applications of cryptography in the security of communications and electronic transactions, telecommunications and data are presented and analyzed. Keywords. Signal, Information, cryptographic protocols, security services Tools used: gp Pari, GPG

Conférence 3 : Les besoins actuels en cryptographie (Lucien Blégban N’GUESSAN)
After a brief history and explaining why cryptography remained secret, we will discuss the basic principles and different encryption systems used, explaining the advantages and disadvantages of each. Through simple examples from everyday life, ranging from the use of the internet, mobile phones, bank cards to electronic signature, but also the technical movement of artisans and even people's cultures, including African, of Electronic signing of documents, we will see how cryptography combines everyday. Faced with the challenge of cryptography, what should be the position of African governments, and how to form cryptographers, respectful of secrets they will manipulate.

Mise en page inspiré du site PARI/GP