En el Marco del Programa de Posgrado, y financiados por el MEC, se impartirán los siguientes cursos y seminarios abiertos a cualquier persona interesada.
Todos los cursos se impartirán en el C-XV-420.
This mini-course gives an introduction to Edwards curves, their fast arithmetic and algebraic properties. We consider Edwards curves over the rationals and over finite fields and rings. The latter is important in the Elliptic Curve Method (ECM) of factorization. ECM is in turn one of the critical tools used in the Number Field Sieve (NFS), the fastest factorization attack against RSA. We will give an overview of factorization methods and describe their combination in the bigger scope of factorizations of RSA numbers.
Background knowledge: some algebra and number theory. Familiarity with finite fields, elliptic curves, and factorization methods is helpful but we try to keep the course self-contained.
+ Seminario: "Complete addition laws for elliptic curves" 17 de Abril
This course is intended for graduate students in the field of cryptography and mathematics.
The content of the course is:
- Discrete logarithm problem (DLP) in generic groups: We present generic methods for computing discrete logarithms
(Babystep Giant step, Pollard's rho and lambda methods, Pohlig Hellman reduction).
- Discrete logarithm in finite fields (Index calculus).
- Cryptographic protocols based on the DLP (Diffie-Hellman key exchange, ElGamal encryption, ...).
- Pairing in cryptography: we introduce identity based cryptography and bilinear pairings.
- Pairing on elliptic curves.
Some examples will be given using SAGE.
+ Seminario: "Rationality of the intersection points of a line with a plane quartic" 27 de Abril
To get a good elliptic cryptosystem, one needs a group with a large prime order. This implies that one has to compute the order of the group of rational points on elliptic curves over finite fields. We will study three major approaches to this problem : the SEA and CM algorithms when the characteristic of the field is big and the AGM method when the characteristic is small.
+ Seminario: "Explicit computation of Serre's obstruction for genus 3 curves" 22 de Mayo
Última Modificación: 22/5/2009