eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-07-17
36:1
36:21
10.4230/LIPIcs.CCC.2020.36
article
Hardness of Bounded Distance Decoding on Lattices in 𝓁_p Norms
Bennett, Huck
1
Peikert, Chris
1
University of Michigan, Ann Arbor, MI, USA
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 𝓁_p norm. We prove that BDD_{p, α} is NP-hard under randomized reductions where α → 1/2 as p → ∞ (and for α = 1/2 when p = ∞), 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 ∈ [1,∞) ⧵ 2ℤ 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 → ∞), 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 > p₁ ≈ 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 𝓁_p norms, which we construct by modifying the integer-lattice sparsification technique of Aggarwal and Stephens-Davidowitz (STOC 2018).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol169-ccc2020/LIPIcs.CCC.2020.36/LIPIcs.CCC.2020.36.pdf
Lattices
Bounded Distance Decoding
NP-hardness
Fine-Grained Complexity