The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations

We prove the following about the Nearest Lattice Vector Problem (in any l_p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is A/P-hard. 2. If for some ε > 0 there exists a polynomial-time algorithm that approximates the optimum within a factor of 2^{log0.5-ε n}, then every NP language can be decided in quasi-polynomial deterministic time, i.e., NP ⊆ DTIME(n^{poly(log n)}). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l_∞ norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2^{log1-ε n} or n^ε. Improving the factor 2^{log0.5-ε n} to √dimension for either of the lattice problems would imply the hardness of the Shortest Vector Problem in l₂ norm; an old open problem. Our proofs use reductions from few-prover, one-round interactive proof systems [FL], BG+], either directly, or through a set-cover problem.