On the hardness of the Learning With Errors problem and its variants

Lattice-based approach to build cryptographic schemes is very promising as we can observe in the ongoing NIST competition for post-quantum cryptography. Indeed, it seems to be an interesting candidate to obtain constructions which resist to attacks using a quantum computer.
Since the work of Short in 1997,  the hardness of number theoretical assumptions, which are used as security foundations for many primitives, is extremely reduced when facing a quantum computer. Even if powerful enough quantum computers do not exist yet, it is necessary to be prepared and anticipate their arrival.
Lattice-based cryptography regroups the approaches which consist in building cryptographic constructions and protocols with their security relying on hard problems on lattices. But we need to use intermediate problem to build such schemes, and the Learning With Errors problem is one of them. In particular, 3 of the 7 NIST finalists have their security relying on one of its variants. In this context, I will present my research work of the last years.
I will first introduce my contribution on studying the hardness of the Learning With Errors problem and its variants. Then, I will give an overview of an other side of my research work which concerns cryptographic constructions and in particular advanced signature schemes.