We prove that SVP$_p$ is NP-hard to approximate within a factor of $2^{\log^{1 - \varepsilon} n}$, for all constants $\varepsilon > 0$ and $p > 2$, under standard deterministic Karp reductions. This result is also the first proof that \emph{exact} SVP$_p$ is NP-hard in a finite $\ell_p$ norm. Hardness for SVP$_p$ with $p$ finite was previously only known if NP $\not \subseteq$ RP, and under that assumption, hardness of approximation was only known for all constant factors. As a corollary to our main theorem, we show that under the Sliding Scale Conjecture, SVP$_p$ is NP-hard to approximate within a small polynomial factor, for all constants $p > 2$. Our proof techniques are surprisingly elementary; we reduce from a \emph{regularized} PCP instance directly to the shortest vector problem by using simple gadgets related to Vandermonde matrices and Hadamard matrices.

翻译：我们证明SVP$_p$对于所有常数$\varepsilon > 0$和$p > 2$，在标准确定性Karp归约下，在因子$2^{\log^{1 - \varepsilon} n}$内近似是NP难的。这一结果也是第一个证明在有限$\ell_p$范数下精确SVP$_p$是NP难的。此前仅已知在NP $\not \subseteq$ RP的条件下SVP$_p$（$p$为有限值）的困难性，且在该假设下已知的近似困难性仅针对所有常数因子。作为我们主定理的推论，我们证明在滑动标度猜想下，对于所有常数$p > 2$，SVP$_p$在多项式小因子内近似是NP难的。我们的证明技术出乎意料地简单；通过使用与Vandermonde矩阵和Hadamard矩阵相关的简单构造，我们直接从正则化PCP实例归约到最短向量问题。