ESPITAU Thomas

PhD student at Sorbonne University
Team : ALMASTY
https://espitau.github.io
https://espitau.github.io

Supervision : Antoine JOUX
Co-supervision : FOUQUE Pierre-Alain

Algorithmic aspects of algebraic lattices

Lattices are mathematical objects generalizing the concrete idea of grid embedded in the plane. They play a fundamental role in the study of various subfields of mathematics and computer science, in particular, algebraic number theory and cryptography. This thesis deals with so-called "algebraic" lattices, that is, constructed above a maximal order of a number field, with a particular emphasis on computational methods. After developing generic techniques enabling the certified manipulation of such objects, we will turn to the development of an effective algorithms for the reduction of lattices over cyclotomic fields, in particular exploiting their natural recursive and symplectic structure. This study is then used for the resolution of a central problem in algorithmic number theory, namely the principal ideal problem, consisting of the finding of a generator a principal ideal in a number field. We eventually look at the implications of these works in public-key cryptography, where we present attacks on a fully homomorphic encryption scheme and on the BLISS digital signature.


Phd defence : 01/14/2020

Jury members :

M. Philippe Elbaz-Vincent , Professeur, Université Grenoble-Alpes [rapporteur]
M. Claus Fieker, Professor Dr., Technische Universitat Kaiserslautern [rapporteur]
Mme. Valérie Berthé, Directrice de Recherche, IRIF
M. Pierre-Alain Fouque, Professeur, Université de Rennes
M. Antoine Joux, Tenured Research Faculty , CISPA, Helmholtz Center
Mme. Ariane Mézard, Professeur, IMJ-PRG, Sorbonne Université
M. Phong Nguyen , Directeur de Recherche, INRIA

Departure date : 01/16/2020

2016-2020 Publications