Problem
Research questions and friction points this paper is trying to address.
Finding short non-zero lattice vectors efficiently
Recursive framework for dense sublattices discovery
Alternative approach to basis reduction algorithms
Recursive lattice reduction - A framework for finding short lattice vectors
🤖 AI Summary
This paper investigates the Shortest Vector Problem (SVP) and the construction of dense sublattices in integer lattices. We propose the first abstract, recursion-based lattice reduction framework that is independent of any specific basis, Gram–Schmidt orthogonalization, or projection operations; instead, it achieves structured approximation via hierarchical search over low-rank sublattices and their duals. Our method unifies SVP solving and the search for arbitrary-rank ℓ-sublattices satisfying min{ℓ, n−ℓ} ≤ n−k+1, yielding improved trade-offs between oracle query complexity and approximation quality. Theoretically, we obtain a quasipolynomial-time algorithm for finding short vectors. Practically, leveraging automated algorithm search, we recover the performance of classical basis reduction algorithms and discover novel, provably superior alternatives.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Recursive framework for finding dense sublattices
Complementary to basis reduction algorithms
Automated search for better approximation algorithms
🔎 Similar Papers
No similar papers found.
