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On the Complexity of Lattice Problems with Polynomial Approximation Factors

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The LLL Algorithm

Part of the book series: Information Security and Cryptography ((ISC))

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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  1. School of Computer Science, Tel-Aviv University, Tel-Aviv, 69978, Israel

    Oded Regev

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  1. Oded Regev
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Editors and Affiliations

  1. Dépt. Mathématiques et d'Informatique, Ecole Normale Supérieure, rue d'Ulm 45, Paris CX 05, 75230, France

    Phong Q. Nguyen

  2. Dept. Informatique, Université de Caen, Caen CX, 14032, France

    Brigitte Vallée

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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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  • DOI: https://doi.org/10.1007/978-3-642-02295-1_15

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02294-4

  • Online ISBN: 978-3-642-02295-1

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