Abstract
Lattice problems are known to be hard to approximate to withinsub-polynomial factors. For larger approximation factors, such as \(\sqrt{n}\), lattice problems are known to be in complexity classes, such as NP ∩ coNP, and are hence unlikely to be NP-hard. Here, we survey known results in this area. We also discuss some related zero-knowledge protocols for lattice problems.
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Acknowledgements
This chapter is partly based on lecture notes scribed by Michael Khanevsky as well as on the paper [24] coauthored with Dorit Aharonov. I thank Ishay Haviv and the anonymous reviewers for their comments on an earlier draft. I also thank Daniele Micciancio for pointing out that the argument in Section “NP-Hardness” extends to the search version. Supported by the Binational Science Foundation, by the Israel Science Foundation, by the European Commission under the Integrated Project QAP funded by the IST directorate as Contract Number 015848, and by a European Research Council (ERC) Starting Grant.
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Regev, O. (2009). On the Complexity of Lattice Problems with Polynomial Approximation Factors. In: Nguyen, P., Vallée, B. (eds) The LLL Algorithm. Information Security and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02295-1_15
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