We propose a recursive lattice reduction framework for finding short non-zero vectors or dense sublattices of a lattice. The framework works by recursively searching for dense sublattices of dense sublattices (or their duals) with progressively lower rank. When the procedure encounters a recursive call on a lattice $L$ with relatively low rank, we simply use a known algorithm to find a shortest non-zero vector in $L$. This new framework is complementary to basis reduction algorithms, which similarly work to reduce an $n$-dimensional lattice problem with some approximation factor $\gamma$ to a lower-dimensional exact lattice problem in some lower dimension $k$, with a tradeoff between $\gamma$, $n$, and $k$. Our framework provides an alternative and arguably simpler perspective. For example, our algorithms can be described at a high level without explicitly referencing any specific basis of the lattice, the Gram-Schmidt orthogonalization, or even projection (though, of course, concrete implementations of algorithms in this framework will likely make use of such things). We present a number of instantiations of our framework. Our main concrete result is an efficient reduction that matches the tradeoff achieved by the best-known basis reduction algorithms. This reduction also can be used to find dense sublattices with any rank $\ell$ satisfying $\min\{\ell,n-\ell\} \leq n-k+1$, using only an oracle for SVP in $k$ dimensions, with slightly better parameters than what was known using basis reduction. We also show a simple reduction with the same tradeoff for finding short vectors in quasipolynomial time, and a reduction from finding dense sublattices of a high-dimensional lattice to this problem in lower dimension. Finally, we present an automated search procedure that finds algorithms in this framework that (provably) achieve better approximations with fewer oracle calls.

翻译：我们提出了一种递归格约化框架，用于寻找格中的短非零向量或稠密子格。该框架通过递归搜索具有逐步降低秩的稠密子格（或其对偶格）的稠密子格来工作。当过程遇到对秩相对较低的格$L$进行递归调用时，我们直接使用已知算法来寻找$L$中的最短非零向量。这一新框架与基约化算法互补，后者类似地将具有近似因子$\gamma$的$n$维格问题约化为某个较低维度$k$中的低维精确格问题，并在$\gamma$、$n$和$k$之间进行权衡。我们的框架提供了一个替代性且可能更简单的视角。例如，我们的算法可以在高层面上描述，而无需显式引用格的任何特定基、Gram-Schmidt正交化，甚至投影（当然，该框架中算法的具体实现可能会使用这些工具）。我们提出了该框架的多种实例化。我们的主要具体结果是一个高效约化，其达到了与最著名基约化算法相同的权衡。该约化也可用于寻找具有任意秩$\ell$的稠密子格，其中$\min\{\ell,n-\ell\} \leq n-k+1$，仅需使用$k$维SVP预言机，且参数略优于已知的基于基约化的方法。我们还展示了一个具有相同权衡的简单约化，用于在拟多项式时间内寻找短向量，以及一个从寻找高维格的稠密子格到低维该问题的约化。最后，我们提出了一种自动化搜索程序，可在该框架中找到（可证明地）以更少预言机调用实现更好近似效果的算法。