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DOI: 10.4230/LIPIcs.CCC.2020.36
URN: urn:nbn:de:0030-drops-125881
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Bennett, Huck ; Peikert, Chris

Hardness of Bounded Distance Decoding on Lattices in ?_p Norms

LIPIcs-CCC-2020-36.pdf (0.6 MB)


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

  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 =		{},
  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}

Keywords: Lattices, Bounded Distance Decoding, NP-hardness, Fine-Grained Complexity
Collection: 35th Computational Complexity Conference (CCC 2020)
Issue Date: 2020
Date of publication: 17.07.2020

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