Lattice problems and norm embeddings

Abstract

We present reductions from lattice problems in the l2 norm to the corresponding problems in other norms such as l1, l∞ (and in fact in any other lp norm where 1 ≤ p ≤ ∞). We consider lattice problems such as the Shortest Vector Problem, Shortest Independent Vector Problem, Closest Vector Problem and the Closest Vector Problem with Preprocessing. Most reductions are simple and follow from known constructions of embeddings of normed spaces.Among other things, our reductions imply that the Shortest Vector Problem in the l1 norm and the Closest Vector Problem with Preprocessing in the l∞ norm are hard to approximate to within any constant (and beyond). Previously, the former problem was known to be hard to approximate to within 2-ε, while no hardness result was known for the latter problem.