For odd
integers pβ₯1 (and p=β), we show that the Closest Vector Problem
in the βpβ 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 pβ₯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 2n+o(n)-time
algorithm is known. In particular, our result applies for any p=p(n)ξ =2
that approaches 2 as nββ.
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 quantitative hardness results for \CVP_p and
\CVPP_p for any 1β€p<β under different assumptions