在$2^n$时间内求解最近向量问题--离散高斯又来了!
Solving the Closest Vector Problem in $2^n$ Time--- The Discrete Gaussian Strikes Again!
摘要：
给出了求解$n$维欧氏格上精确最近向量问题（CVP）的$2^{n+o(n)}$-时空随机算法.这改进了以前最快的算法，确定性的$\widetilde{O}(4^{n})$-时间和$\widetilde{O}(2^{n})$-Micciancio和Voulgaris的空间算法。我们通过三个步骤来实现我们的主要结果。首先，我们展示了如何修改[ADRS 15]中的采样算法，以解决具有非常低参数的格点移位$L- t$上的离散高斯采样问题。虽然实际算法是[ADRS 15]的自然推广，但分析使用了大量新思想。这产生了用于任何近似因子$γ= 1+2^{-o(n/\log n)}$的近似CVP的$2^{n+o(n)}$时间算法。第二，我们表明，近似最接近的向量的目标向量$t$可以被分组为"低维集群"，我们使用它来获得递归减少从确切的CVP的一个变体的近似CVP，"表现良好，这些集群。"第三，我们证明了我们的离散高斯采样算法可以用来解决近似CVP的这种变体。分析关键取决于离散高斯分布和近似最近的向量，这可能是独立的利益的一些新的属性。
英文摘要：
We give a $2^{n+o(n)}$-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on $n$-dimensional Euclidean lattices. This improves on the previous fastest algorithm, the deterministic $\widetilde{O}(4^{n})$-time and $\widetilde{O}(2^{n})$-space algorithm of Micciancio and Voulgaris. We achieve our main result in three steps. First, we show how to modify the sampling algorithm from [ADRS15] to solve the problem of discrete Gaussian sampling over lattice shifts, $L- t$, with very low parameters. While the actual algorithm is a natural generalization of [ADRS15], the analysis uses substantial new ideas. This yields a $2^{n+o(n)}$-time algorithm for approximate CVP for any approximation factor $γ= 1+2^{-o(n/\log n)}$. Second, we show that the approximate closest vectors to a target vector $t$ can be grouped into "lower-dimensional clusters," and we use this to obtain a recursive reduction from exact CVP to a variant of approximate CVP that "behaves well with these clusters." Third, we show that our discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP. The analysis depends crucially on some new properties of the discrete Gaussian distribution and approximate closest vectors, which might be of independent interest.
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