Short Bases of Lattices over Number Fields

Lattices over number elds arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of Z-lattices, the choice of a nice, short (pseudo)-basis is important in many applications. In this article, we provide the rst algorithm that computes such a short (pseudo)-basis. We utilize the LLL algorithm for Z-lattices together with the Bosma-Pohst-Cohen Hermite Normal Form and some size reduction technique to nd a pseudo-basis where each basis vector belongs to the lattice and the product of the norms of the basis vectors is bounded by the lattice determinant, up to a multiplicative factor that is a eld invariant. As it runs in polynomial time, this provides an e ective variant of Minkowski's second theorem for lattices over number elds.