The objectives of this work package are to provide theoretical and practical foundations for work packages 4, 5 and 6. On the theoretical front, we need to extend existing models and techniques from the classical setting to the quantum one. On the practical front, we need to explore improvements of lattice trapdoors, which constitute the efficiency bottleneck in many applications. We will also consider cryptanalytic aspects, with three main questions. First, we need to determine whether structured lattices (which greatly improve performances) are also quantum-secure. Second, we wish to improve our understanding of the concrete security of lattice cryptosystems, so as to derive practical parameters for future instantiations. Finally, we will work on the resistance of implementation of lattice-based cryptography to side-channel attacks.