The inapproximability of lattice and coding problems with preprocessing

Abstract

We prove that the closest vector problem with preprocessing (CVPP) is NP-hard to approximate within any factor less than 5 3 . More specifically, we show that there exists a reduction from an NP-hard problem to the approximate closest vector problem such that the lattice depends only on the size of the original problem, and the specific instance is encoded solely in the target vector. It follows that there are lattices for which the closest vector problem cannot be approximated within factors γ< 5 3 in polynomial time, no matter how the lattice is represented, unless NP is equal to P (or NP is contained in P/poly, in case of nonuniform sequences of lattices). The result easily extends to any ℓ p norm, for p⩾1, showing that CVPP in the ℓ p norm is hard to approximate within any factor γ< 5 3 p . As an intermediate step, we establish analogous results for the nearest codeword problem with preprocessing (NCPP), proving that for any finite field GF(q), NCPP over GF( q) is NP-hard to approximate within any factor less than 5 3 .