Emplacement
Dates
Présentation
The aim of this school is to introduce students to some aspect of (algorithmic) number theory and arithmetic geometry and the very fruitful interplay between those subjects and the applied disciplines of cryptography and coding theory. Our program consists of four courses on algorithmic number theory, elliptic curves, algebraic coding theory and isogeny based cryptography, respectively. These courses will introduce the students to a variety of tools in number theory and arithmetic geometry as well as their applications in cryptography and coding theory. Topics will include class groups of number fields and Buchmann's subexponential algorithm for computing them, the Mordell-Weil theorem for elliptic curves over number fields, Reed-Muller code and Goppa code, algorithms to compute and evaluate isogenies of elliptic curves. Every course will combine theoretical and practical aspects: exercises and programming sessions will be held in all four courses, putting students' proactive learning at the center of this project.
Le programme scientifique est disponible sur le site local de l'Ecole : http://www.rnta.eu/HCMC2024/
Langue officielle de l'Ecole : Anglais
Coordinateurs administratifs et scientifiques
Programme scientifique
Cours 1: "introductory course - Weierstrass Equations, The Group Law, Projective Space and the Point at Infinity, Cubic Equations, Quartic Equations, The j-invariant Elliptic Curves in Characteristic 2, Endomorphisms, Singular Curves, Torsion Points, Division Polynomials, The Weil Pairing, Elliptic Curves over Finite Fields, The Frobenius Endomorphism, Supersingular Curves, Elliptic Curves over Q, The Torsion Subgroup. The Lutz-Nagell Theorem, the Mordell-Weil Theorem. Elliptic curves over the complex numbers. Elliptic curves with complex multiplication. The main theorem of complex multiplication.Supersingular curves and maximal orders of quaternion algebras.", Francesco PAPPALARDI (Università Roma Tre, Italy)
Cours 2: "advanced course - In this course, we will first introduce the class group and the unit group of a number field and study the structures of these groups. After that, we study algorithms to compute these for arbitrary number fields. In particular, we describe Buchmann’s subexponential algorithm. In this context we introduce the ideal lattices associated to a number field and the equivalent concept of Arakelov divisors. We define the Arakelov class group and the concept of a reduced Arakelov divisor. Computing short vectors in ideal lattices is the key ingredient of Buchmann’s algorithm In addition, special attention will be paid to algorithms to compute class groups of imaginary quadratic fields. Here the Arakelov divisors are simply 2-dimensional lattices in C. Classically the theory and the algorithms are described in terms of binary quadratic forms. This part will be used in the security analysis of isogeny-based cryptography.", Laura GEATTI (Università di Roma "Tor Vergata", Italy)
Cours 3: "introductory course - Algebraic coding theory rests on arithmetic and finite fields. The first part of this course will be devoted to this background. The theory of cyclotomic polynomials is a basic tool. The second part will start with the definitions and basic properties of codes, many examples will be given, including Hamming codes, Golay codes, Bose-Chaudhuri- Hocquenghem codes, Reed-Solomon codes. Many exercices will be proposed. 1. Background: Arithmetic, finite fields. Cyclic groups Residue classes modulo n The ring $\mathbb{Z}[X]$ M ̈obius inversion formula Gauss fields Cyclotomic polynomials Decomposition of cyclotomic polynomials over a finite field Trace and Norm Infinite Galois theory 2. Coding Theory Some historical dates Hamming distance Definitions, Examples Cyclic codes Detection, correction and minimal distance Hamming codes Generator matrix and check matrix Minimum distance of a code Golay codes Duality and self-duality Sphere packing and Singleton bounds Maximum distance separable codes Perfect codes Weight enumerator Reed-Mueller codes Goppa codes Generalized Hamming distance Rank distance Generalized rank distance", Michel WALDSCHMIDT (Sorbonne Université, France)
Cours 5: "exercise session - The student will present their solution of exercises assigned during classes.", Valerio TALAMANCA (Università Roma Tre, Italy)
Cours 6: "exercise session - The student will present their solution of exercises assigned during classes.", Laura GEATTI (Università di Roma "Tor Vergata", Italy)
Cours 7: "programming sessions - The student will use PARI to implement some of the algorithm introduced during the lectures and made explicit computations.", Laura GEATTI (Università di Roma "Tor Vergata", Italy)
Cours 8: "exercise session - The student will present their solution of exercises assigned during classes.", Michel WALDSCHMIDT (Sorbonne Université, France)
Cours 10: "programming sessions - The goal is to familiarize everyone with the functions of PARI commonly used when dealing with elliptic curves and perform some explict calculation for specific examples.", Francesco PAPPALARDI (Università Roma Tre, Italy)
Cours 11: "programming sessions - The programming session will use SAGE and will give an overview of what can be done with the library and how to use the main functionalities.", Frederique Elise OGGIER (Nanyang Technological University, Singapore)
Site internet de l'école
Comment participer
Pour s'inscrire et postuler à un financement CIMPA, suivre les instructions données ici https://www.cimpa.info/en/node/40
Date limite d'inscription : février 28, 2024