AI 中文总结
本文提出局部稠密格问题LDLP，证明特定ℓ_p范数下该问题是多项式谱系第二层完全问题，同时验证了两种局部密度标准定义的鲁棒等价性。
AI 中文摘要
局部稠密格是用于证明最短向量问题及相关格问题难解性的核心构造工具。非正式地说，局部稠密格是指这样的格$\u2112$：在某个以$\u2192{s}$为球心、半径至多为其最短非零格向量长度的$α<1$倍的$\u2113_p$球内，包含指数级数量的格向量。\n 本文从“元”视角研究局部稠密格，提出了局部稠密格问题（Locally Dense Lattice Problem，LDLP），即判定给定输入是否对应一个局部稠密格的决策问题。我们的主要结果是：对于所有满足$p \geq \log_2 3$的有限$p$值的$\u2113_p$范数以及无穷范数下的LDLP，都是多项式谱系第二层的完全问题。\n 我们还比较了已有研究中出现的两种局部密度的标准定义。Micciancio最初的定义（FOCS 1998和SICOMP 2001）使用整数系数向量，而Micciancio后续工作（ToC 2012）以及Bennett和Peikert的工作（RANDOM 2023）则使用平移陪集中的短向量。我们证明，对应的两个承诺问题可在确定性多项式时间内互相归约，表明这两种表述形式具有鲁棒等价性。
英文摘要
\emph{Locally dense lattices} are central gadgets used to prove the hardness of the Shortest Vector Problem and related lattice problems. Informally, a locally dense lattice is a lattice $\mathcal{L}$ that contains exponentially many lattice vectors inside some $\ell_p$ ball centered at $\vec{s}$ with radius at most an $α< 1$ fraction of the length of its shortest nonzero lattice vector. In this paper, taking a ``meta'' viewpoint on locally dense lattices, we introduce the \emph{Locally Dense Lattice Problem} (LDLP), the decision problem of determining whether a given input specifies a locally dense lattice. Our main result is that LDLP in $\ell_p$ norms for all finite $p \geq \log_2 3$ and for the infinity norm is complete for the second level of the polynomial hierarchy. We also compare two standard definitions of local density that appear in prior work. Micciancio's original definition (FOCS 1998 and SICOMP 2001) uses integer coefficient vectors, while later work by Micciancio (ToC 2012) and by Bennett and Peikert (RANDOM 2023) uses short vectors in a shifted coset. We show that the corresponding promise problems are mutually reducible in deterministic polynomial time, which shows that the two formulations are robust.
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