The DOI matches its Crossref record.
Hardness of approximating the shortest vector problem in lattices
Abstract
Let p > 1 be any fixed real. We show that assuming NP ⊈ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in ℓ p norm within a constant factor. Under the stronger assumption NP ⊈ RTIME(2 poly (log n ) ), we show that there is no polynomial-time algorithm with approximation ratio 2 (log n ) 1/2−ϵ where n is the dimension of the lattice and ϵ > 0 is an arbitrarily small constant.We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product , a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2 (log n ) 1/2-ϵ .
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cited by
Worst‐Case to Average‐Case Reductions Based on Gaussian Measures
Micciancio, D., & Regev, O. (2007) · SIAM Journal on Computing · doi:10.1137/s0097539705447360 · Cited by 907
Regev, O. (2006) · Lecture Notes in Computer Science · doi:10.1007/11818175_8 · Cited by 485
Quantum algorithms for algebraic problems
Childs, A. M., & van Dam, W. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.1 · Cited by 344
Private quantum computation: an introduction to blind quantum computing and related protocols
Fitzsimons, J. F. (2017) · npj Quantum Information · doi:10.1038/s41534-017-0025-3 · Cited by 295
A Decade of Lattice Cryptography
Peikert, C. (2016) · now publishers, Inc. eBooks · doi:10.1561/9781680831139 · Cited by 256
Metadata
- Authors
- Khot, S.
- Venue
- Journal of the ACM
- Published
- 2005-09
- Volume
- 52
- Issue
- 5
- Pages
- 789-808
- ISSN
- 0004-5411 (ISSN-L)
- 1557-735X
- OpenAlex
- W2056492141
- Cited by
- 212
- References
- 0
- FWCI
- 7.22
- Type
- journal article
- Language
- en
- Open access
- closed
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: Hardness of approximating the shortest vector problem in lattices. https://app.cochat.ai/papers/10.1145/1089023.1089027
- The paper
- Khot, S. (2005). Hardness of approximating the shortest vector problem in lattices. Journal of the ACM. https://doi.org/10.1145/1089023.1089027
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
