Principles of Lattice Cryptography, and cryptanalysis by lattice reduction

In this presentation, I will present lattice-based cryptography as stemming from tessellating a Euclidean vector space according using a lattice basis. This directly points at lattice reduction algorithm for cryptanalysis, and I will cover the famous LLL algorithm, and discuss stronger but slower algorithm.
If times allow, I will also discuss the special case of ideal lattices of a number field, how the reduction theory differs, and how it can be exploited for attacks and hardness proof.