We propose a new framework called recursive lattice reduction for finding short non-zero vectors in a lattice or for finding dense sublattices of a lattice. At a high level, the framework works by recursively searching for dense sublattices of dense sublattices (or their duals). Eventually, the procedure encounters a recursive call on a lattice $\mathcal{L}$ with relatively low rank $k$, at which point we simply use a known algorithm to find a short non-zero vector in $\mathcal{L}$. We view our framework as complementary to basis reduction algorithms, which similarly work to reduce an $n$-dimensional lattice problem with some approximation factor $\gamma$ to an exact lattice problem in dimension $k < n$, with a tradeoff between $\gamma$, $n$, and $k$. Our framework provides an alternative and arguably simpler perspective, which in particular can be described without explicitly referencing any specific basis of the lattice, Gram-Schmidt vectors, or even projection (though implementations of algorithms in this framework will likely make use of such things). We present a number of specific instantiations of our framework. Our main concrete result is a reduction that matches the tradeoff between $\gamma$, $n$, and $k$ achieved by the best-known basis reduction algorithms (in terms of the Hermite factor, up to low-order terms) across all parameter regimes. In fact, 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 (or even just Hermite SVP) in $k$ dimensions, which is itself a novel result (as far as the authors know). We also show a very simple reduction that achieves the same tradeoff in quasipolynomial time. Finally, we present an automated approach for searching for algorithms in this framework that (provably) achieve better approximations with fewer oracle calls.

翻译：我们提出一种名为递归格约简的新框架，用于在格中寻找非零短向量或寻找格中稠密子格。从高层来看，该框架通过递归搜索稠密子格的稠密子格（或其对偶格）来运行。最终，该过程会遇到对低秩 $k$ 的格 $\mathcal{L}$ 的递归调用，此时我们直接使用已知算法来寻找 $\mathcal{L}$ 中的非零短向量。我们将此框架视为基约简算法的补充，后者类似地将具有近似因子 $\gamma$ 的 $n$ 维格问题归约为维度 $k < n$ 的精确格问题，并在 $\gamma$、$n$ 和 $k$ 之间进行权衡。我们的框架提供了一种替代且可论证更简单的视角，特别是它可以不明确引用格的具体基、Gram-Schmidt 向量甚至投影来描述（尽管该框架中算法的实现很可能利用这些工具）。我们展示该框架的若干具体实例化。主要具体结果是一种归约方法，在所有参数范围内均匹配已知最佳基约简算法（以埃尔米特因子计，忽略低阶项）在 $\gamma$、$n$ 和 $k$ 之间实现的权衡。事实上，该归约还可用于寻找任意秩 $\ell$（满足 $\min\{\ell,n-\ell\} \leq n-k+1$）的稠密子格，仅需使用 $k$ 维 SVP（甚至仅埃尔米特 SVP）预言机，这本身即为一项新成果（据作者所知）。我们还展示了一种极为简单的归约，可在拟多项式时间内实现相同权衡。最后，我们提出一种自动化方法，用于在该框架内搜索（可证明）以更少预言机调用实现更好近似的算法。