Hardness of Bounded Distance Decoding on Lattices in đ_p Norms
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| Titel: | Hardness of Bounded Distance Decoding on Lattices in đ_p Norms |
|---|---|
| Autoren: | Bennett, Huck, Peikert, Chris |
| Weitere Verfasser: | Huck Bennett and Chris Peikert |
| Verlagsinformationen: | Schloss Dagstuhl â Leibniz-Zentrum fĂŒr Informatik |
| Publikationsjahr: | 2020 |
| Bestand: | DROPS - Dagstuhl Research Online Publication Server (Schloss Dagstuhl - Leibniz Center for Informatics ) |
| Schlagwörter: | Lattices, Bounded Distance Decoding, NP-hardness, Fine-Grained Complexity |
| Beschreibung: | 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 đ_p norm. We prove that BDD_{p, α} is NP-hard under randomized reductions where α â 1/2 as p â â (and for α = 1/2 when p = â), 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 â [1,â) ⧔ 2†and constants C > 1, Δ > 0, there is no 2^((1-Δ)n/C)-time algorithm for BDD_{p,α} for some constant α (which approaches 1/2 as p â â), 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 > pâ â 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 đ_p norms, which we construct by modifying the integer-lattice sparsification technique of Aggarwal and Stephens-Davidowitz (STOC 2018). |
| Publikationsart: | article in journal/newspaper conference object |
| Dateibeschreibung: | application/pdf |
| Sprache: | 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 |
| VerfĂŒgbarkeit: | 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 |
| Dokumentencode: | edsbas.34C2C4F6 |
| Datenbank: | BASE |
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| Items | – Name: Title Label: Title Group: Ti Data: Hardness of Bounded Distance Decoding on Lattices in 𝓁_p Norms – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Bennett%2C+Huck%22">Bennett, Huck</searchLink><br /><searchLink fieldCode="AR" term="%22Peikert%2C+Chris%22">Peikert, Chris</searchLink> – Name: Author Label: Contributors Group: Au Data: Huck Bennett and Chris Peikert – Name: Publisher Label: Publisher Information Group: PubInfo Data: Schloss Dagstuhl â Leibniz-Zentrum für Informatik – Name: DatePubCY Label: Publication Year Group: Date Data: 2020 – Name: Subset Label: Collection Group: HoldingsInfo Data: DROPS - Dagstuhl Research Online Publication Server (Schloss Dagstuhl - Leibniz Center for Informatics ) – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Lattices%22">Lattices</searchLink><br /><searchLink fieldCode="DE" term="%22Bounded+Distance+Decoding%22">Bounded Distance Decoding</searchLink><br /><searchLink fieldCode="DE" term="%22NP-hardness%22">NP-hardness</searchLink><br /><searchLink fieldCode="DE" term="%22Fine-Grained+Complexity%22">Fine-Grained Complexity</searchLink> – Name: Abstract Label: Description Group: Ab Data: 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 â 1/2 as p â â (and for α = 1/2 when p = â), 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 â [1,â) ⧔ 2†and constants C > 1, Δ > 0, there is no 2^((1-Δ)n/C)-time algorithm for BDD_{p,α} for some constant α (which approaches 1/2 as p â â), 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 > pâ â 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). – Name: TypeDocument Label: Document Type Group: TypDoc Data: article in journal/newspaper<br />conference object – Name: Format Label: File Description Group: SrcInfo Data: application/pdf – Name: Language Label: Language Group: Lang Data: English – Name: NoteTitleSource Label: Relation Group: SrcInfo Data: Is Part Of LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020); https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.36 – Name: DOI Label: DOI Group: ID Data: 10.4230/LIPIcs.CCC.2020.36 – Name: URL Label: Availability Group: URL Data: https://doi.org/10.4230/LIPIcs.CCC.2020.36<br />https://nbn-resolving.org/urn:nbn:de:0030-drops-125881<br />https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.36 – Name: Copyright Label: Rights Group: Cpyrght Data: https://creativecommons.org/licenses/by/3.0/legalcode – Name: AN Label: Accession Number Group: ID Data: edsbas.34C2C4F6 |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.4230/LIPIcs.CCC.2020.36 Languages: – Text: English Subjects: – SubjectFull: Lattices Type: general – SubjectFull: Bounded Distance Decoding Type: general – SubjectFull: NP-hardness Type: general – SubjectFull: Fine-Grained Complexity Type: general Titles: – TitleFull: Hardness of Bounded Distance Decoding on Lattices in đ_p Norms Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Bennett, Huck – PersonEntity: Name: NameFull: Peikert, Chris – PersonEntity: Name: NameFull: Huck Bennett and Chris Peikert IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Type: published Y: 2020 Identifiers: – Type: issn-locals Value: edsbas – Type: issn-locals Value: edsbas.oa |
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