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 [7], Goldreich and Goldwasser [14], and Aharonov and Regev [2]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series 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 [26]. An interesting fact is that our result emerged from a “dequantization” of our previous quantum result in [2]. This route to proving purely classical results might be beneficial elsewhere.