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Hardness of Bounded Distance Decoding on Lattices in 𝓁_p Norms

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Title: Hardness of Bounded Distance Decoding on Lattices in 𝓁_p Norms
Authors: Bennett, Huck, Peikert, Chris
Contributors: Huck Bennett and Chris Peikert
Publisher Information: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Publication Year: 2020
Collection: DROPS - Dagstuhl Research Online Publication Server (Schloss Dagstuhl - Leibniz Center for Informatics )
Subject Terms: Lattices, Bounded Distance Decoding, NP-hardness, Fine-Grained Complexity
Description: 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).
Document Type: article in journal/newspaper
conference object
File Description: application/pdf
Language: English
Relation: Is Part Of LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020); https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.36
DOI: 10.4230/LIPIcs.CCC.2020.36
Availability: https://doi.org/10.4230/LIPIcs.CCC.2020.36
https://nbn-resolving.org/urn:nbn:de:0030-drops-125881
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.36
Rights: https://creativecommons.org/licenses/by/3.0/legalcode
Accession Number: edsbas.34C2C4F6
Database: BASE
Description
Abstract: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).
DOI:10.4230/LIPIcs.CCC.2020.36