On the Quantitative Hardness of CVP

For odd integers p ≥ 1 (and p = ∞), we show that the Closest Vector Problem in the ℓ _p norm (CVP _p ) over rank n lattices cannot be solved in 2 ^(1-ε)n time for any constant ε > 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 <; ∞ under different assumptions.