Recursive lattice reduction - A framework for finding short lattice vectors.

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.