We show that, assuming NP $\not\subseteq$ $\cap_{δ> 0}$DTIME$\left(\exp{n^δ}\right)$, the shortest vector problem for lattices of rank $n$ in any finite $\ell_p$ norm is hard to approximate within a factor of $2^{(\log n)^{1 - o(1)}}$, via a deterministic reduction. Previously, for the Euclidean case $p=2$, even hardness of the exact shortest vector problem was not known under a deterministic reduction.

翻译：我们证明，假设NP $\not\subseteq$ $\cap_{δ> 0}$DTIME$\left(\exp{n^δ}\right)$，在任意有限$\ell_p$范数中，秩为$n$的格的最短向量问题在因子$2^{(\log n)^{1 - o(1)}}$内难以近似，且该结论通过确定性归约得到。此前，对于欧几里得情形$p=2$，即使在确定性归约下，精确最短向量问题的难度也尚未可知。