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The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant
Abstract
We show that approximating the shortest vector problem (in any $\ell_p$ norm) to within any constant factor less than $\sqrt[p]2$ is hard for NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (random polynomial time), unless NP equals RP. We also prove a proper NP-hardness result (i.e., hardness under deterministic many-one reductions) under a reasonable number theoretic conjecture on the distribution of square-free smooth numbers. As part of our proof, we give an alternative construction of Ajtai's constructive variant of Sauer's lemma that greatly simplifies Ajtai's original proof.
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Cited by
Efficient Fully Homomorphic Encryption from (Standard) LWE
Brakerski, Z., & Vaikuntanathan, V. (2011) · 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science · doi:10.1109/focs.2011.12 · Cited by 1,449
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
Self-citationMicciancio, D., & Regev, O. (2009) · Post-Quantum Cryptography · doi:10.1007/978-3-540-88702-7_5 · Cited by 615
Self-citationQuantum 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
Metadata
- Authors
- Micciancio, D.
- Venue
- SIAM Journal on Computing
- Published
- 2001-01
- Volume
- 30
- Issue
- 6
- Pages
- 2008-2035
- ISSN
- 0097-5397 (ISSN-L)
- 1095-7111
- OpenAlex
- W2144593761
- Cited by
- 212
- References
- 0
- FWCI
- 9.09
- Type
- journal article
- Language
- en
- Open access
- closed
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