**Algebraic Geometry, Number Theory and Applications in Cryptography and Robot kinematics .**

**Location: **AIMS-Cameroon, Limbe.

**Presentation of the School**

This school aims to offer an intensive teaching session to graduate students and young researchers. That concerns key topics in Algebraic Geometry and Number Theory. Indeed many classical results and methods in these areas are used in flourishing domains of applied mathematics. We selected the following 6 courses: basic algebraic number theory and class field theory, Tate module and Abelian varieties, quantitative and algorithmic recent results in real algebraic geometry, advanced topics in semi-algebraic geometry, counting points on algebraic varieties, and fundamental groups in Algebraic and Arithmetic Geometry. These fundamental courses describe all theoretical elements needed for the applications in cryptography and robot kinematics which will be developed at the end of the school. Beyond lectures, we are also planning sessions devoted to solving exercises and to computer experiments with Pari/GP and Sage. We expect that at the end of the school every participant will be able to select a suitable hyperelliptic curve $C$ for constructing some cryptosystems based on the discrete logarithm problem in its Jacobian $J_C$.

**Scientific Committee :**

- Marie-Françoise Roy, Université de Rennes 1, marie-francoise.roy@univ-rennes1.fr
- Michel Coste, Université de Rennes 1, michel.coste@univ-rennes1.fr
- Christian Maire, Université de Besançon, christian.maire@univ-fcomte.fr
- Marco Andrea Garuti, Universita Degli Studi Di Padova, mgaruti@math.unipd.it

**Organizing committee :**

- Mama Foupouagnigni, AIMS-Cameroon, mfoupouagnigni@aims-cameroon.org
- Aminatou Pecha, University of Maroua, aminap2001@yahoo.fr
- Marie-Françoise Ouedraogo, Université de Ouagadougou, omfrancoise@yahoo.fr
- Mary Fomboh, University of Buea, fomboh_mary@yahoo.com

**Registration. Click here to register**

**Important Dates :**

**Dates of School :**July 2-13, 2019**Deadline for registration of CIMPA participants :**March 24, 2019

**Scientific program**- Course 1 :
**Basic algebraic number theory and class field theory**

Lecturer: Elisa Lorenzo, Université de Rennes 1

**Abstract.** *We will start by studying the structure of the decomposition of prime ideals in number fields and by discussing the definitions of norm, trace and discriminant. From there we will move to the basics of Class Field Theory: we will define the Artin symbol and we will state the Reciprocity Law. We will end by showing the applications of the Class Field Theory to the Theory of the Complex Multiplication. All the course will be illustrated with several examples which will help to the understanding of these deep theories.*

- Course 2 :
**Tate Module and Abelian Varieties**

Lecturer : Christian Maire, Université de Besançon

**Abstract**. *In this course, we will introduce the key concepts (and some basic tools) of Galois representations of Tate modules of Abelian varieties (elliptic curves and more generally Jacobian varieties).*

* We will first spend a certain time on elliptic curves to introduce in detail some notions in order to well understand their Tate module: locus of ramification, Frobenius and characteristic polynomial, mod p representation, L-function, image of the representation, modularity, etc.*

* After that, we will explain how these properties extend to the the case of genus >1.*

- Course 3 :
**Quantitative and algorithmic recent results in real algebraic geometry**

Lecturer : Marie-Françoise Roy, Université de Rennes 1

**Abstract**. *Important theoretical results in real algebraic geometry such as the algebraic proofs of the fundamental theorem of algebra (valid for a real closed field), the curve selection lemma, the finiteness theorem (i.e a closed semi-algebraic set has closed description) have been recently studied from a quantitative and algorithmic point of view. Several methods are used: the cylindrical decomposition and the critical point method. In both cases, algebraic results about sub-resultants play an important role.*

- Course 4 :
**Advanced topics in semi-algebraic geometry and modelization in Robot Kinematics**

Lecturer: Michel Coste, Université de Rennes 1

**Abstract**. *The course will give a short introduction to Robot Kinematics and show examples of applications of algebraic and semialgebraic geometry in this field. I shall discuss direct and inverse kinematics and singularities, especially for parallel robots. I shall also discuss mechanisms having several operating modes, with possibly different degrees of freedom. I shall explain methods to translate problems of robot kinematics into systems of polynomial equations, including the model of the group of rigid motions given by the Study quadric, using dual quaternions. The effective methods of algebraic and semialgebraic geometry can then be applied (elimination, decomposition into primary components, cylindrical algebraic decomposition …). Problems to study with the help of computer algebra systems will be given to the students.*

- Course 5 :
**Point counting on algebraic varieties and applications in cryptography**

Lecturer : Tony Ezome, Université des Sciences et Techniques de Masuku

**Abstract**. *Given an algebraic variety $V$ over a finite field $F_q$, we know that the $F_{q^k}$-rational points on $V$ form a finite set. What arises naturally in our mind is the construction of a process which computes the number of $F_{q^k}$-rational points in $V$. This is one of the most important and very recurrent questions in cryptography, particularly when $V$ is a (hyper-)elliptic curve $C$ or a Jacobian variety $J_C$. That led to many points counting algorithms. This course aims to describe the more important methods. We will start with the naive algorithm (enumeration of points) which is a quite general method, and then we will describe the Baby Step Giant Step algorithm for elliptic curves. We will explain how are related the Frobenius endomorphism of a curve $C$, the number of rational points on $C$, the number of rational points on the Jacobian $J_C$, and the Weil Conjectures. We will also describe the Schoof $l$-adic algorithm and the main steps in SEA algorithm. We will end by giving a technique for selecting a hyperelliptic curve $C$ (and the underlying finite field) suitable for implementing a discrete logarithm cryptosystem in the Jacobian variety $J_C$.*

- Course 6:
**Fundamental groups in Algebraic and Arithmetic Geometry**

Lecturer: Marco Garuti, Universita Degli Studi di Padova

**Abstract**. *The course is a survey on the theory of Fundamental Groups in Algebraic and Arithmetic Geometry. Starting from Grothendieck’s theory developed in SGA 1, we will review his Anabelian philosophy and its applications to the search for points on varieties*.

**Contacts :**Dr Aminatou Pecha,

** Tél.** (+237) 6 77 40 53 62 – aminap2001@yahoo.fr

**Sponsors **

- PRMAIS
- AIMS Cameroon
- IMU
- French laboratories