The inapproximability of lattice and coding problems with preprocessing

We prove that the closest vector problem with preprocessing (CVPP) is NP-hard to approximate within an any factor less than root5/3. 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 tat-get vector It follows that there are lattices for which the closest vector problem cannot be approximated within factors gamma < root5/3 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 gamma < root5/3. 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.