  $ \newcommand{\SVP}{\mathsf{SVP}} \newcommand{\NP}{\mathsf{NP}}
\newcommand{\RTIME}{\mathsf{RTIME}} \newcommand{\RSUBEXP}{\mathsf{RSUBEXP}}
\newcommand{\eps}{\epsilon} \newcommand{\poly}{\mathop{\mathrm{poly}}} $We show
that unless $\NP \subseteq \RTIME (2^{\poly(\log{n})})$, there is no
polynomial-time algorithm approximating the Shortest Vector Problem ($\SVP$) on
$n$-dimensional lattices in the $\ell_p$ norm ($1 \leq p< \infty$) to within a
factor of $2^{(\log{n})^{1-\eps}}$ for any $\eps > 0$. This improves the
previous best factor of $2^{(\log{n})^{1/2-\eps}}$ under the same complexity
assumption due to Khot (J. ACM, 2005). Under the stronger assumption $\NP
\nsubseteq \RSUBEXP$, we obtain a hardness factor of $n^{c/\log\log{n}}$ for
some $c> 0$.
  Our proof starts with Khot's $\SVP$ instances that are hard to approximate to
within some constant. To boost the hardness factor we simply apply the standard
tensor product of lattices. The main novelty is in the analysis, where we show
that the lattices of Khot behave nicely under tensorization. At the heart of
the analysis is a certain matrix inequality which was first used in the context
of lattices by de Shalit and Parzanchevski (2006).
