30/10, IMPA. Sala 228
Charles Favre (École Polytechnique, Centre de Mathématiques Laurent Schwartz),
Non-Archimedean links of normal surface singularities.|
Resumo. (j.w. with Lorenzo Fantini and Matteo Ruggiero) One can associate to any complex normal surface singularity a non-Archimedean analog of its classical link. This non-Archimedean link carries a natural analytic structure that is locally modelled on Berkovich analytic spaces over C((t)). We shall explain how to obtain a characterization of sandwich singularities in term of self-similar properties of this link.
Ben Smith (École Polytechnique, INRIA),
Applications of arithmetic geometry in contemporary cryptology.|
Resumo. Elliptic curves and low-dimensional Jacobian varieties over finite fields are an important tool in contemporary public-key cryptography.
Formally---that is, from the point of view of cryptographic protocols---they can often be used as a drop-in replacement for the multiplicative group of a finite field, where they offer higher levels of security with much more compact keys. But this purely formal point of view ignores the rich arithmetic structure of elliptic curves, Jacobians, and their Kummer varieties, which we can exploit to create practical improvements (and even some cryptographic attacks) that have no analogues in conventional finite field-based cryptosystems.
In this talk, we will survey some explicit applications of the arithmetic of low-dimensional abelian varieties in cryptography. These techniques include
* the construction of models with more efficient group laws and scalar multiplication operations,
* using isogenies and endomorphisms to accelerate encryption and decryption algorithms,
* using endomorphism ring structures to accelerate point counting (ie, zeta function computation) for elliptic and genus 2 curves, and
* using isogenies to attack discrete logarithm problems.