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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2020.36
URN: urn:nbn:de:0030-drops-125881
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12588/
Bennett, Huck ;
Peikert, Chris
Hardness of Bounded Distance Decoding on Lattices in ?_p Norms
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 ?_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 ?_p norms, which we construct by modifying the integer-lattice sparsification technique of Aggarwal and Stephens-Davidowitz (STOC 2018).
BibTeX - Entry
@InProceedings{bennett_et_al:LIPIcs:2020:12588,
author = {Huck Bennett and Chris Peikert},
title = {{Hardness of Bounded Distance Decoding on Lattices in ?_p Norms}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {36:1--36:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Shubhangi Saraf},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12588},
URN = {urn:nbn:de:0030-drops-125881},
doi = {10.4230/LIPIcs.CCC.2020.36},
annote = {Keywords: Lattices, Bounded Distance Decoding, NP-hardness, Fine-Grained Complexity}
}
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Keywords: |
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Lattices, Bounded Distance Decoding, NP-hardness, Fine-Grained Complexity |
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Collection: |
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35th Computational Complexity Conference (CCC 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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17.07.2020 |
