On the Quantitative Hardness of CVP

newcommandepsvarepsilon newcommandproblem[1]ensuremath#1 newcommandCVPproblemCVP newcommandSVPproblemSVP newcommandCVPPproblemCVPP newcommandensuremath[1]#1 For odd integers p ≥ 1 (and p = ∞), we show that the Closest Vector Problem in the ℓp norm (CVPp) 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 ≥ 1, not including the even integers. This comes tantalizingly close to settling the quantitative time complexity of the important special case of CVP2 (i.e., CVP in the Euclidean norm), for which a 2n +o(n)-time algorithm is known. In particular, our result applies for any p = p(n) neq 2 that approaches 2 as n → ∞. We also show a similar SETH-hardness result for SVP_∞; hardness of approximating CVPp to within some constant factor under the so-called Gap-ETH assumption; and other quantitative hardness results for CVPp and CVPPp for any 1 ≤ p < ∞ under different assumptions.