On Bounded Distance Decoding for General Lattices

Abstract

A central problem in the algorithmic study of lattices is the closest vector problem: given a lattice $\mathcal{L}$ represented by some basis, and a target point $\vec{y}$, find the lattice point closest to $\vec{y}$. Bounded Distance Decoding is a variant of this problem in which the target is guaranteed to be close to the lattice, relative to the minimum distance $\lambda_1(\mathcal{L})$ of the lattice. Specifically, in the α-Bounded Distance Decoding problem (α-BDD), we are given a lattice $\mathcal{L}$ and a vector $\vec{y}$ (within distance $\alpha\cdot\lambda_1(\mathcal{L})$ from the lattice), and we are asked to find a lattice point $\vec{x}\in \mathcal{L}$ within distance $\alpha\cdot\lambda_1(\mathcal{L})$ from the target. In coding theory, the lattice points correspond to codewords, and the target points correspond to lattice points being perturbed by noise vectors. Since in coding theory the lattice is usually fixed, we may “pre-process” it before receiving any targets, to make the subsequent decoding faster. This leads us to consider α-BDD with pre-processing. We show how a recent technique of Aharonov and Regev [2] can be used to solve α-BDD with pre-processing in polynomial time for $\alpha=O\left(\sqrt{(\log{n})/n}\right)$. This improves upon the previously best known algorithm due to Klein [13] which solved the problem for $\alpha=O\left(1/n\right)$. We also establish hardness results for α-BDD and α-BDD with pre-processing, as well as generalize our results to other ℓ _p norms.