Hardness of Bounded Distance Decoding on Lattices in 𝓁_p Norms

Authors Huck Bennett, Chris Peikert

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Huck Bennett
  • University of Michigan, Ann Arbor, MI, USA
Chris Peikert
  • University of Michigan, Ann Arbor, MI, USA


We thank the Simons Institute for hosting the Spring 2020 program "Lattices: Algorithms, Complexity, and Cryptography," at which some of this work was completed. We also thank Noah Stephens-Davidowitz for sharing his plot-generating code from Aggarwal and Stephens-Davidowitz (STOC 2018) with us.

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Huck Bennett and Chris Peikert. Hardness of Bounded Distance Decoding on Lattices in 𝓁_p Norms. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Lattices
  • Bounded Distance Decoding
  • NP-hardness
  • Fine-Grained Complexity


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