不同$p$ -范数下SVP和CVP之间的保偏约化
Dimension-Preserving Reductions Between SVP and CVP in Different $p$-Norms
摘要：
$ \newcommand{\SVP}{\textsf{SVP}} \newcommand{\CVP}{\textsf{CVP}} \newcommand{\eps}{\varepsilon} $我们显示了一些减少之间的最短向量问题和最接近的向量问题的格在不同的$\ell_p$规范（$\SVP_p$和$\CVP_p$分别）。具体地，我们给出了以下$1 \leq p \leq q \leq \infty$的$2^{\eps m}$ -时间约简，它们都将输入格的秩$n$和维数$m$增加最多1：$\bullet$-从$\widetilde{O}(1/\eps^{1/p})γ$ -近似$\SVP_q$到$γ$ -近似$\SVP_p$的约简; $\bullet$-从$\widetilde{O}(1/\eps^{1/p}) γ$ -近似$\CVP_p$到$γ$ -近似$\CVP_q$的约简;而$\bullet$是从$\widetilde{O}(1/\eps^{1+1/p})$ - $\CVP_q$到$(1+\eps)$ -唯一$\SVP_p$的约简（其又平凡地约简到$(1+\eps)$ -近似$\SVP_p$）。即使在$p = q$的情况下，最后的减少也是有趣的。特别是，这种特殊的情况下，包括许多以前的工作适应$2^{O(m)}$时间$\SVP_p$算法来解决$O(1)$近似$\CVP_p$。在$p = q$、$1 \leq p \leq 2$和$\SVP_p$ oracle是精确的（重要的）特殊情况下，我们在$2^{\eps m}$时间内显示了从$O(1/\eps^{1/p})\text{-}\CVP_p$到（精确的）$\SVP_p$的更强的约简。例如，取$\eps = \log m/m$和$p = 2$，比Kannan著名的多项式时间约简（从$\sqrt{m}\text{-}\CVP_2$到$\SVP_2$）有轻微的改进。我们还注意到，最后两个约简可以结合起来，对于任何$p$和$q$，无论是$p \leq q$还是$p > q$，都可以从近似- $\CVP_p$约简到$\SVP_q$。我们的技术结合了Eisenbrand和Venzin最近的突破性工作（展示了如何将$\ell_2$规范中这些问题的当前最快已知算法适应于所有$\ell_p$规范）以及基于稀疏化的技术。
英文摘要：
$ \newcommand{\SVP}{\textsf{SVP}} \newcommand{\CVP}{\textsf{CVP}} \newcommand{\eps}{\varepsilon} $We show a number of reductions between the Shortest Vector Problem and the Closest Vector Problem over lattices in different $\ell_p$ norms ($\SVP_p$ and $\CVP_p$ respectively). Specifically, we present the following $2^{\eps m}$-time reductions for $1 \leq p \leq q \leq \infty$, which all increase the rank $n$ and dimension $m$ of the input lattice by at most one: $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p})γ$-approximate $\SVP_q$ to $γ$-approximate $\SVP_p$; $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p}) γ$-approximate $\CVP_p$ to $γ$-approximate $\CVP_q$; and $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1+1/p})$-$\CVP_q$ to $(1+\eps)$-unique $\SVP_p$ (which in turn trivially reduces to $(1+\eps)$-approximate $\SVP_p$). The last reduction is interesting even in the case $p = q$. In particular, this special case subsumes much prior work adapting $2^{O(m)}$-time $\SVP_p$ algorithms to solve $O(1)$-approximate $\CVP_p$. In the (important) special case when $p = q$, $1 \leq p \leq 2$, and the $\SVP_p$ oracle is exact, we show a stronger reduction, from $O(1/\eps^{1/p})\text{-}\CVP_p$ to (exact) $\SVP_p$ in $2^{\eps m}$ time. For example, taking $\eps = \log m/m$ and $p = 2$ gives a slight improvement over Kannan's celebrated polynomial-time reduction from $\sqrt{m}\text{-}\CVP_2$ to $\SVP_2$. We also note that the last two reductions can be combined to give a reduction from approximate-$\CVP_p$ to $\SVP_q$ for any $p$ and $q$, regardless of whether $p \leq q$ or $p > q$. Our techniques combine those from the recent breakthrough work of Eisenbrand and Venzin (which showed how to adapt the current fastest known algorithm for these problems in the $\ell_2$ norm to all $\ell_p$ norms) together with sparsification-based techniques.
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