$\newcommand{\Z}{\mathbb{Z}}$ We show improved fine-grained hardness of two key lattice problems in the $\ell_p$ norm: Bounded Distance Decoding to within an $\alpha$ factor of the minimum distance ($\mathrm{BDD}_{p, \alpha}$) and the (decisional) $\gamma$-approximate Shortest Vector Problem ($\mathrm{SVP}_{p,\gamma}$), assuming variants of the Gap (Strong) Exponential Time Hypothesis (Gap-(S)ETH). Specifically, we show: 1. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for $\mathrm{BDD}_{p, \alpha}$ for any constant $\alpha > \alpha_\mathsf{kn}$, where $\alpha_\mathsf{kn} = 2^{-c_\mathsf{kn}} < 0.98491$ and $c_\mathsf{kn}$ is the $\ell_2$ kissing-number constant, unless non-uniform Gap-ETH is false. 2. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for $\mathrm{BDD}_{p, \alpha}$ for any constant $\alpha > \alpha^\ddagger_p$, where $\alpha^\ddagger_p$ is explicit and satisfies $\alpha^\ddagger_p = 1$ for $1 \leq p \leq 2$, $\alpha^\ddagger_p < 1$ for all $p > 2$, and $\alpha^\ddagger_p \to 1/2$ as $p \to \infty$, unless randomized Gap-ETH is false. 3. For all $p \in [1, \infty) \setminus 2 \Z$, all $C > 1$, and all $\varepsilon > 0$, there is no $2^{(1-\varepsilon)n/C}$-time algorithm for $\mathrm{BDD}_{p, \alpha}$ for any constant $\alpha > \alpha^\dagger_{p, C}$, where $\alpha^\dagger_{p, C}$ is explicit and satisfies $\alpha^\dagger_{p, C} \to 1$ as $C \to \infty$ for any fixed $p \in [1, \infty)$, unless non-uniform Gap-SETH is false. 4. For all $p > p_0 \approx 2.1397$, $p \notin 2\Z$, and all $\varepsilon > 0$, there is no $2^{(1-\varepsilon)n/C_p}$-time algorithm for $\mathrm{SVP}_{p, \gamma}$ for some constant $\gamma = \gamma(p, \varepsilon) > 1$ and explicit constant $C_p > 0$ where $C_p \to 1$ as $p \to \infty$, unless randomized Gap-SETH is false.

中文翻译：

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提高 BDD 和 SVP 在 Gap-(S)ETH 下的硬度

$\newcommand{\Z}{\mathbb{Z}}$ 我们在 $\ell_p$ 范数中展示了两个关键晶格问题的改进细粒度硬度：有界距离解码到最小距离的 $\alpha$ 因子内($\mathrm{BDD}_{p, \alpha}$) 和（决定性的）$\gamma$-近似最短向量问题（$\mathrm{SVP}_{p,\gamma}$），假设差距（强）指数时间假设（差距-（S）ETH）。具体来说，我们证明： 1. 对于所有 $p \in [1, \infty)$，$\mathrm{BDD}_{p, \alpha} 没有 $2^{o(n)}$-time 算法$ 对于任何常数 $\alpha > \alpha_\mathsf{kn}$，其中 $\alpha_\mathsf{kn} = 2^{-c_\mathsf{kn}} < 0.98491$ 和 $c_\mathsf{kn}$是 $\ell_2$ 接吻数常数，除非非统一 Gap-ETH 为假。2. 对于所有的 $p \in [1, \infty)$，$\mathrm{BDD}_{p 没有 $2^{o(n)}$-time 算法，\alpha}$ 对于任何常数 $\alpha > \alpha^\ddagger_p$，其中 $\alpha^\ddagger_p$ 是显式的并且满足 $\alpha^\ddagger_p = 1$ for $1 \leq p \leq 2$, $ \alpha^\ddagger_p < 1$ 对于所有 $p > 2$，以及 $\alpha^\ddagger_p \to 1/2$ 作为 $p \to \infty$，除非随机化的 Gap-ETH 是假的。3. 对于所有 $p \in [1, \infty) \setminus 2 \Z$，所有 $C > 1$，以及所有 $\varepsilon > 0$，没有 $2^{(1-\varepsilon)n /C}$-time 算法 $\mathrm{BDD}_{p, \alpha}$ 对于任何常数 $\alpha > \alpha^\dagger_{p, C}$，其中 $\alpha^\dagger_{p , C}$ 是明确的并且满足 $\alpha^\dagger_{p, C} \to 1$ as $C \to \infty$ 对于任何固定的 $p \in [1, \infty)$，除非不均匀Gap-SETH 是假的。4. 对于所有 $p > p_0 \approx 2.1397$、$p \notin 2\Z$ 和所有 $\varepsilon > 0$，