Research
Introduction
Information theory has become a critical field due to its key role in the
advance of global communication networks as well as everyday technology items
such as smart cards, storage media, and much more.
Over the past decades, its development has been largely enabled by the use of
deep mathematical results and objects, mostly originating from the vast field
of algebraic geometry, such as elliptic curves.
Although such objects have been studied from a pure mathematical standpoint for
centuries, their effective and computational aspects, without which
applications to information theory would not exist, have only been investigated
for a few decades and remain largely unexplored.
The GAATI Laboratory brings together mathematicians working on topics of great
significance to information theory, and directs its research effort towards:
- unlocking the effective aspects of those topics;
- exploiting them to further the state-of-the-art in information theory.
Topics
Although not limited to the list below, the current research interests of the
GAATI Laboratory include:
-
Boolean functions of small differential uniformity:
- APN functions and the APN conjecture;
- construction of functions of small differential uniformity;
- applications to the construction of block ciphers.
-
Computational aspects of algebraic curves:
- Jacobian varieties and the efficient computation of their group law;
- point-counting algorithms and the construction of secure curves;
- construction of efficient curve-based cryptographic systems;
- constructive and destructive applications of pairings in cryptography;
- effective aspects of complex multiplication theory and its applications to cryptography.
-
Algebraic function fields and their computational aspects:
- generalizing existing constructions of towers of extensions, in particular
to the case of Kummer extensions and of Carlitz-Hayes cyclotomic extensions;
- applications to the design of faster multiplication algorithms for finite fields;
- construction of algebraic curves defined on finite fields with many rational points.
