Hardness of Bounded Distance Decoding on Lattices in π_p Norms
Huck Bennett, Chris Peikert Β· DROPS (Schloss Dagstuhl β Leibniz Center for Informatics) Β· 2020
Bounded Distance Decoding BDD_{p,Ξ±} is the problem of decoding a lattice when the target point is promised to be within an Ξ± factor of the minimum distance of the lattice, in the π_p norm. We prove that BDD_{p, Ξ±} is NP-hard under randomized reductions where Ξ± β 1/2 as p β β (and for Ξ± = 1/2 when p = β), 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,Ξ±}. For example, we prove that for all p β [1,β) ⧡ 2β€ and constants C > 1, Ξ΅ > 0, there is no 2^((1-Ξ΅)n/C)-time algorithm for BDD_{p,Ξ±} for some constant Ξ± (which approaches 1/2 as p β β), 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,Ξ±} by Liu, Lyubashevsky, and Micciancio (APPROX-RANDOM 2008), our results improve the values of Ξ± for which the problem is known to be NP-hard for all p > pβ β 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 π_p norms, which we construct by modifying the integer-lattice sparsification technique of Aggarwal and Stephens-Davidowitz (STOC 2018).