求平均值！SVP（和CVP）的一个非常简单的$2^n$ -时间算法
Just Take the Average! An Embarrassingly Simple $2^n$-Time Algorithm for SVP (and CVP)
摘要：
我们展示了一个$2^{n+o(n)}$ -时间（和空间）算法的最短向量问题的格子（SVP），其工作原理是通过重复运行一个非常简单的“对和平均”筛样程序的一个列表上的格子向量。这与Aggarwal，Dadush，Regev和Stephens-Davidowitz（ADRS，STOC，2015）的当前最快已知算法的运行时间（和空间）相匹配，具有更简单的算法。我们的算法实际上是一个修改的ADRS算法，与一定的谨慎拒绝采样步骤删除。我们算法的正确性来自一个更一般的“元定理”，表明这种拒绝采样步骤对于某类算法和用例是不必要的。特别是，这也适用于Aggarwal，Dadush和Stephens-Davidowitz（ADS，FOCS，2015）的最近向量问题（CVP）的相关$2^{n + o(n)}$ -时间算法，为任何$γ= 1+2^{-o(n/\log n)}$的$γ$ -近似CVP提供了类似的简单算法。(We也可以从用于精确CVP的$2^{n+o(n)}$ -时间ADS算法中去除拒绝采样过程，但是得到的算法仍然相当复杂。）
英文摘要：
We show a $2^{n+o(n)}$-time (and space) algorithm for the Shortest Vector Problem on lattices (SVP) that works by repeatedly running an embarrassingly simple "pair and average" sieving-like procedure on a list of lattice vectors. This matches the running time (and space) of the current fastest known algorithm, due to Aggarwal, Dadush, Regev, and Stephens-Davidowitz (ADRS, in STOC, 2015), with a far simpler algorithm. Our algorithm is in fact a modification of the ADRS algorithm, with a certain careful rejection sampling step removed. The correctness of our algorithm follows from a more general "meta-theorem," showing that such rejection sampling steps are unnecessary for a certain class of algorithms and use cases. In particular, this also applies to the related $2^{n + o(n)}$-time algorithm for the Closest Vector Problem (CVP), due to Aggarwal, Dadush, and Stephens-Davidowitz (ADS, in FOCS, 2015), yielding a similar embarrassingly simple algorithm for $γ$-approximate CVP for any $γ= 1+2^{-o(n/\log n)}$. (We can also remove the rejection sampling procedure from the $2^{n+o(n)}$-time ADS algorithm for exact CVP, but the resulting algorithm is still quite complicated.)
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