  $ \newcommand{\SVP}{\textsf{SVP}} \newcommand{\CVP}{\textsf{CVP}}
\newcommand{\eps}{\varepsilon} $We show a number of reductions between the
Shortest Vector Problem and the Closest Vector Problem over lattices in
different $\ell_p$ norms ($\SVP_p$ and $\CVP_p$ respectively). Specifically, we
present the following $2^{\eps m}$-time reductions for $1 \leq p \leq q \leq
\infty$, which all increase the rank $n$ and dimension $m$ of the input lattice
by at most one:
  $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p})\gamma$-approximate
$\SVP_q$ to $\gamma$-approximate $\SVP_p$;
  $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p}) \gamma$-approximate
$\CVP_p$ to $\gamma$-approximate $\CVP_q$; and
  $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1+1/p})$-$\CVP_q$ to
$(1+\eps)$-unique $\SVP_p$ (which in turn trivially reduces to
$(1+\eps)$-approximate $\SVP_p$).
  The last reduction is interesting even in the case $p = q$. In particular,
this special case subsumes much prior work adapting $2^{O(m)}$-time $\SVP_p$
algorithms to solve $O(1)$-approximate $\CVP_p$. In the (important) special
case when $p = q$, $1 \leq p \leq 2$, and the $\SVP_p$ oracle is exact, we show
a stronger reduction, from $O(1/\eps^{1/p})\text{-}\CVP_p$ to (exact) $\SVP_p$
in $2^{\eps m}$ time. For example, taking $\eps = \log m/m$ and $p = 2$ gives a
slight improvement over Kannan's celebrated polynomial-time reduction from
$\sqrt{m}\text{-}\CVP_2$ to $\SVP_2$. We also note that the last two reductions
can be combined to give a reduction from approximate-$\CVP_p$ to $\SVP_q$ for
any $p$ and $q$, regardless of whether $p \leq q$ or $p > q$.
  Our techniques combine those from the recent breakthrough work of Eisenbrand
and Venzin (which showed how to adapt the current fastest known algorithm for
these problems in the $\ell_2$ norm to all $\ell_p$ norms) together with
sparsification-based techniques.
