  $ \newcommand{\eps}{\varepsilon}
\newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} }
\newcommand{\CVP}{\problem{CVP}} \newcommand{\SVP}{\problem{SVP}}
\newcommand{\CVPP}{\problem{CVPP}} \newcommand{\ensuremath}[1]{#1} $For odd
integers $p \geq 1$ (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. In particular, our result applies for any $p = p(n) \neq 2$
that approaches $2$ as $n \to \infty$.
  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 $1 \leq p < \infty$ under different assumptions.
