We establish deterministic hardness of approximation results for the Shortest Vector Problem in $\ell_p$ norm ($\mathsf{SVP}_p$) and for Unique-SVP ($\mathsf{uSVP}_p$) for all $p > 2$. Previously, no deterministic hardness results were known, except for $\ell_\infty$. For every $p > 2$, we prove constant-ratio hardness: no polynomial-time algorithm approximates $\mathsf{SVP}_p$ or $\mathsf{uSVP}_p$ within a ratio of $\sqrt{2} - o(1)$, assuming $\textsf{3SAT} \notin \text{DTIME}(2^{O(n^{2/3}\log n)})$, and, $\textsf{Unambiguous-3SAT} \notin \text{DTIME}(2^{O(n^{2/3}\log n)})$. We also show that for any $\varepsilon > 0$ there exists $p_\varepsilon > 2$ such that for every $p \ge p_\varepsilon$: no polynomial-time algorithm approximates $\mathsf{SVP}_p$ within a ratio of $2^{(\log n)^{1- \varepsilon}}$, assuming $\text{NP} \nsubseteq \text{DTIME}(n^{(\log n)^\varepsilon})$; and within a ratio of $n^{1/(\log\log(n))^\varepsilon}$, assuming $\text{NP} \nsubseteq \text{SUBEXP}$. This improves upon [Haviv, Regev, Theory of Computing 2012], which obtained similar inapproximation ratios under randomized reductions. We obtain analogous results for $\mathsf{uSVP}_p$ under the assumptions $\textsf{Unambiguous-3SAT} \not\subseteq \text{DTIME}(n^{(\log n)^\varepsilon})$ and $\textsf{Unambiguous-3SAT} \not\subseteq \text{SUBEXP}$, improving the previously known $1+o(1)$ [Stephens-Davidowitz, Approx 2016]. Strengthening the hardness of $\textsf{uSVP}$ has direct cryptographic impact. By the reduction of Lyubashevsky and Micciancio [Lyubashevsky, Micciancio, CRYPTO 2009], hardness for $\gamma$-$\mathsf{uSVP}_p$ carries over to ${\frac{1}{\gamma}}$-$\mathsf{BDD}_p$ (Bounded Distance Decoding). Thus, understanding the hardness of $\textsf{uSVP}$ improves worst-case guarantees for two core problems that underpin security in lattice-based cryptography.

翻译：本文针对所有 $p > 2$ 的情况，建立了 $\ell_p$ 范数下最短向量问题（$\mathsf{SVP}_p$）与唯一最短向量问题（$\mathsf{uSVP}_p$）的确定性近似强度结果。此前除 $\ell_\infty$ 范数外，尚未有确定性强度结论。对于任意 $p > 2$，我们证明了常数倍近似强度：假设 $\textsf{3SAT} \notin \text{DTIME}(2^{O(n^{2/3}\log n)})$ 且 $\textsf{Unambiguous-3SAT} \notin \text{DTIME}(2^{O(n^{2/3}\log n)})$，则不存在多项式时间算法能以 $\sqrt{2} - o(1)$ 的近似比求解 $\mathsf{SVP}_p$ 或 $\mathsf{uSVP}_p$。同时我们证明：对于任意 $\varepsilon > 0$，存在 $p_\varepsilon > 2$，使得对所有 $p \ge p_\varepsilon$：在假设 $\text{NP} \nsubseteq \text{DTIME}(n^{(\log n)^\varepsilon})$ 下，不存在多项式时间算法能以 $2^{(\log n)^{1- \varepsilon}}$ 的近似比求解 $\mathsf{SVP}_p$；在假设 $\text{NP} \nsubseteq \text{SUBEXP}$ 下，不存在多项式时间算法能以 $n^{1/(\log\log(n))^\varepsilon}$ 的近似比求解 $\mathsf{SVP}_p$。该结果改进了 [Haviv, Regev, Theory of Computing 2012] 在随机归约下获得的不可近似比。我们在假设 $\textsf{Unambiguous-3SAT} \not\subseteq \text{DTIME}(n^{(\log n)^\varepsilon})$ 与 $\textsf{Unambiguous-3SAT} \not\subseteq \text{SUBEXP}$ 下，对 $\mathsf{uSVP}_p$ 获得了类似结论，将先前已知的 $1+o(1)$ 近似比结果 [Stephens-Davidowitz, Approx 2016] 向前推进。强化 $\textsf{uSVP}$ 的强度结论对密码学具有直接影响。根据 Lyubashevsky 与 Micciancio 的归约 [Lyubashevsky, Micciancio, CRYPTO 2009]，$\gamma$-$\mathsf{uSVP}_p$ 的强度可转化为 ${\frac{1}{\gamma}}$-$\mathsf{BDD}_p$（有界距离解码问题）的强度。因此，理解 $\textsf{uSVP}$ 的强度可改进格密码学安全基础中两个核心问题的最坏情况保证。