On the Quantitative Hardness of CVP

Abstract

For odd integers p ≥ 1 (and p = ∞), we show that the Closest Vector Problem in the ℓp norm (CVP_p) over rank n lattices cannot be solved in 2^(1-≥) n time for any constant ≥ 0 unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to almost all values of p ≥ 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. In particular, our result applies for any p = p(n) ≠ 2 that approaches 2 as n ↦ ∞.We also show a similar SETH-hardness result for SVP_∞; 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