The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, is organizing a  School in Computational Algebra and Number Theory, to be held in Montevideo, Uruguay, from 5 –  10 December 2014.

Facultad de Ingeniería - Universidad de la República
Julio Herrera y Reissig 565, 11300, Montevideo, Uruguay
 

This school is a satellite of FoCM'14
 

TOPICS AND FACULTY

Teresa Krick (Universidad de Buenos Aires)

Arithmetic Nullstellensätze and Applications 

The purpose of these lectures is to introduce Hilbert's Nullstellensatz, a cornerstone in Algebraic Geometry, and comment on its effective aspects, including degree and height aspects when the defining field admits a notion of height. We will introduce the ingredients used to obtain sharp estimates, which

can be useful on their own, and present some applications (by others) to problems on finite fields. 
 

Christophe Ritzenthaler (Université Rennes 1)

Elliptic Curves and its Applications to Cryptography 

The course will be a brief introduction to elliptic curves and its applications to cryptography and is divided into 4 lectures: 
 

- definition(s) of an elliptic curve, group law, isomorphisms, torsion points, Weil pairing. 

- elliptic curves over finite fields: two proofs of Hasse-Weil bound. 

- elliptic curves over finite fields: how to count points? Overview of the different methods. 

- application to cryptography: protocols, attacks and zoology of the existing models. 

Peter Stevenhagen (Universiteit Leiden) 

Algebraic Number Theory 

- roots of the topic: Fermat, Euler, quadratic reciprocity. 

- algorithmic requirements for the theory, e.g. in view of the number field sieve. 

- working in number rings: integrality, (explicit) ideal factorization 

- geometry of numbers, finiteness theorems (class group, Dirichlet unit theorem). 

- Dedekind zeta function, computing fundamental number field invariants, 

- local-global aspects: local fields, adeles, ideles. 

- number field sieve.