Noah Stephens-Davidowitz, Princeton University

Abstract

For odd integers p >= 1 (and p = infty), we show that the Closest Vector Problem in the ell_p norm (CVP_p) over rank n lattices cannot be solved in 2^{(1-eps) n} time for any constant eps > 0 unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to ``almost all'' values of p geq 1, not including the even integers. This comes tantalizingly close to settling the quantitative time complexity of the important special case of CVP_2 (i.e., CVP in the Euclidean norm), for which a 2^{n +o(n)}-time algorithm is known.

We also show a similar SETH-hardness result for SVP_infty; hardness of approximating CVP_p to within some constant factor under the so-called Gap-ETH assumption; and other hardness results for CVP_p and CVPP_p for any 1 <= p < infty under different assumptions.