Bounded Distance Decoding for Random Lattices

TL;DR

This paper studies the bounded distance decoding problem for lattices with sub-Gaussian generator matrices, proving NP-hardness in the worst case but providing a polynomial-time algorithm that works well on average.

Contribution

It demonstrates NP-hardness of BDD for these lattices and introduces a SVD-based polynomial-time algorithm that succeeds on average, bridging worst-case hardness and average-case efficiency.

Findings

NP-hardness of BDD in worst case for these lattices

A polynomial-time SVD-based algorithm solves BDD on average

First example of a lattice problem NP-hard worst case but efficiently solvable on average

Abstract

The current paper investigates the bounded distance decoding (BDD) problem for ensembles of lattices whose generator matrices have sub-Gaussian entries. We first prove that, for these ensembles the BDD problem is NP-hard in the worst case. Then, we introduce a polynomial-time algorithm based on singular value decomposition (SVD) and establish, both theoretically and through extensive experiments, that, for a random selected lattice from the same ensemble, the algorithm solves the BDD problem with high probability. To the best of our knowledge, this work provides the first example of a lattice problem that is NP-hard in the worst case yet admits a polynomial time algorithm on the average case.