Gap-(S)ETH假设下BDD与SVP的改进硬度结果
Improved Hardness of BDD and SVP Under Gap-(S)ETH
AI总结：
该研究在Gap-(S)ETH变体假设下，证明了ℓ_p范数下BDD与SVP问题的更紧细粒度硬度下界，给出了不同参数范围下的算法时间复杂度不可行性结果。
摘要：
我们在Gap（强）指数时间假设（Gap-(S)ETH）的变体假设下，证明了ℓ_p范数下两个核心格问题的更精细的改进硬度：最小距离α因子内的有界距离解码（$\text{BDD}_{p, α}$），以及（判定性）γ近似最短向量问题（$\text{SVP}_{p,γ}$）。具体结果如下：\n1. 对所有$p \u0026#10006; [1, \u0026#8734;)$，假设$c_\u0026#10084;\u0026#10082;\u0026#10089; \u0026gt; 0$且非均匀Gap-ETH成立，则对任意常数$α\u0026gt; α_\u0026#10084;\u0026#10082;\u0026#10089;$，$\text{BDD}_{p, α}$不存在$2^{o(n)}$时间算法；其中$α_\u0026#10084;\u0026#10082;\u0026#10089; = 2^{-c_\u0026#10084;\u0026#10082;\u0026#10089;}$，$c_\u0026#10084;\u0026#10082;\u0026#10089;$为ℓ_2接吻数常数。\n2. 对所有$p \u0026#10006; [1, \u0026#8734;)$，除非随机化Gap-ETH不成立，否则对任意常数$α\u0026gt; α^\u0026Dagger;_p$，$\text{BDD}_{p, α}$不存在$2^{o(n)}$时间算法；其中$α^\u0026Dagger;_p$为显式常数，满足当$1 \u0026#10877; p \u0026#10877; 2$时$α^\u0026Dagger;_p = 1$，对所有$p \u0026gt; 2$有$α^\u0026Dagger;_p \u0026lt; 1$，且当$p \u0026rarr; \u0026#8734;$时$α^\u0026Dagger;_p \u0026rarr; 1/2$。\n3. 对所有$p \u0026#10006; [1, \u0026#8734;) \u0026#11080; 2\u0026#8484;$以及所有$C \u0026gt; 1$，假设$c_\u0026#10084;\u0026#10082;\u0026#10089; \u0026gt; 0$且非均匀Gap-SETH成立，则对任意常数$α\u0026gt; α^\u0026Dagger;_{p, C}$，$\text{BDD}_{p, α}$不存在$2^{n/C}$时间算法；其中$α^\u0026Dagger;_{p, C}$为显式常数，且对任意固定的$p \u0026#10006; [1, \u0026#8734;)$，当$C \u0026rarr; \u0026#8734;$时$α^\u0026Dagger;_{p, C} \u0026rarr; 1$。\n4. 对所有$p \u0026gt; p_0 \u0026asymp; 2.1397$且$p \u0026#8713; 2\u0026#8484;$，以及所有$C \u0026gt; C_p$，除非随机化Gap-SETH不成立，否则对某个常数$γ\u0026gt; 1$，$\text{SVP}_{p, γ}$不存在$2^{n/C}$时间算法；其中$C_p \u0026gt; 1$为显式常数，且当$p \u0026rarr; \u0026#8734;$时$C_p \u0026rarr; 1$。
英文摘要：
We show improved fine-grained hardness of two key lattice problems in the $\ell_p$ norm: Bounded Distance Decoding to within an $α$ factor of the minimum distance ($\mathrm{BDD}_{p, α}$) and the (decisional) $γ$-approximate Shortest Vector Problem ($\mathrm{SVP}_{p,γ}$), 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 $\mathrm{BDD}_{p, α}$ for any constant $α> α_\mathsf{kn}$, where $α_\mathsf{kn} = 2^{-c_\mathsf{kn}}$ and $c_\mathsf{kn}$ is the $\ell_2$ kissing-number constant, assuming $c_\mathsf{kn} > 0$ and that non-uniform Gap-ETH holds. 2. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for $\mathrm{BDD}_{p, α}$ for any constant $α> α^\ddagger_p$, where $α^\ddagger_p$ is explicit and satisfies $α^\ddagger_p = 1$ for $1 \leq p \leq 2$, $α^\ddagger_p < 1$ for all $p > 2$, and $α^\ddagger_p \to 1/2$ as $p \to \infty$, unless randomized Gap-ETH is false. 3. For all $p \in [1, \infty) \setminus 2 \mathbb{Z}$ and all $C > 1$, there is no $2^{n/C}$-time algorithm for $\mathrm{BDD}_{p, α}$ for any constant $α> α^\dagger_{p, C}$, where $α^\dagger_{p, C}$ is explicit and satisfies $α^\dagger_{p, C} \to 1$ as $C \to \infty$ for any fixed $p \in [1, \infty)$, assuming $c_\mathsf{kn} > 0$ and that non-uniform Gap-SETH holds. 4. For all $p > p_0 \approx 2.1397$, $p \notin 2\mathbb{Z}$, and all $C > C_p$, there is no $2^{n/C}$-time algorithm for $\mathrm{SVP}_{p, γ}$ for some constant $γ> 1$, where $C_p > 1$ is explicit and satisfies $C_p \to 1$ as $p \to \infty$, unless randomized Gap-SETH is false.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
