The hardness of the closest vector problem with preprocessing

We give a new simple proof of the NP-hardness of the closest vector problem. In addition to being much simpler than all previously known proofs, the new proof yields new interesting results about the complexity of the closest vector problem with preprocessing. This is a variant of the closest vector problem in which the lattice is specified in advance, and can be preprocessed for an arbitrarily long amount of time before the target vector is revealed. We show that there are lattices for which the closest vector problem remains hard, regardless of the amount of preprocessing.