Lattice problems in NP ∩ coNP

We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk (1993), Goldreich and Goldwasser (2000), and Aharonov and Regev (2003). Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier transform over the lattice. This technique might be useful elsewhere - we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP,) improving on a previous result of Regev (2003). An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in (Aharanov and Regev, 2003). This route to proving purely classical results might be beneficial elsewhere.