摘要：Bounded Distance Decoding BDD_{p,α} is the problem of decoding a lattice when the target point is promised to be within an α factor of the minimum distance of the lattice, in the ð"_p norm. We prove that BDD_{p, α} is NP-hard under randomized reductions where α â' 1/2 as p â' â^Z (and for α = 1/2 when p = â^Z), thereby showing the hardness of decoding for distances approaching the unique-decoding radius for large p. We also show fine-grained hardness for BDD_{p,α}. For example, we prove that for all p â^^ [1,â^Z) ⧵ 2â"¤ and constants C > 1, ε > 0, there is no 2^((1-ε)n/C)-time algorithm for BDD_{p,α} for some constant α (which approaches 1/2 as p â' â^Z), assuming the randomized Strong Exponential Time Hypothesis (SETH). Moreover, essentially all of our results also hold (under analogous non-uniform assumptions) for BDD with preprocessing, in which unbounded precomputation can be applied to the lattice before the target is available. Compared to prior work on the hardness of BDD_{p,α} by Liu, Lyubashevsky, and Micciancio (APPROX-RANDOM 2008), our results improve the values of α for which the problem is known to be NP-hard for all p > pâ, â^ 4.2773, and give the very first fine-grained hardness for BDD (in any norm). Our reductions rely on a special family of "locally dense" lattices in ð"_p norms, which we construct by modifying the integer-lattice sparsification technique of Aggarwal and Stephens-Davidowitz (STOC 2018).
