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Approximating Closest Vector Problem in \(\ell _\infty \) Norm Revisited

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Algorithmic Aspects in Information and Management (AAIM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11640))

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Abstract

The security of most lattice-based cryptography schemes are based on two computational hard problems which are the Short Integer Solution (SIS) and Learning With Errors (LWE) problems. The computational complexity of SIS and LWE problems are related to approximating Shortest Vector Problem (SVP) and Bounded Distance Decoding Problem (BDD). Approximating BDD is a special case of approximating Closest Vector Problem (CVP).

In this paper, we revisit the study for approximating Closest Vector Problem. We give one proof that approximating the Closest Vector Problem over \(\ell _\infty \) norm (\(\hbox {CVP}_\infty \)) within any constant factor is NP-hard. The result is obtained by the gap-preserving reduction from Min Total Label Cover problem in \(\ell _1\) norm to \(\hbox {CVP}_\infty \). This proof is simpler than known proofs.

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Acknowledgments

We would like to thank the anonymous referees for their careful readings of the manuscripts and many useful suggestions.

Wenbin Chen’s research has been supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11271097., and by the Program for Innovative Research Team in Education Department of Guangdong Province Under No. 2016KCXTD017. Jianer Chen has been supported by the National Natural Science Foundation of China (NSFC) under Grant No. 61872097.

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Authors and Affiliations

  1. School of Computer Science and Cyber Engineering, Guangzhou University, Guangzhou, People’s Republic of China

    Wenbin Chen & Jianer Chen

  2. Department of Computer Science and Engineering, Texas A&M University, College Station, TX, 77843, USA

    Jianer Chen

Authors
  1. Wenbin Chen
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  2. Jianer Chen
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Correspondence to Wenbin Chen .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Hefei University of Technology, Hefei, China

    Lian Li

  3. Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

    Xiaoming Sun

  4. Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

    Jialin Zhang

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Chen, W., Chen, J. (2019). Approximating Closest Vector Problem in \(\ell _\infty \) Norm Revisited. In: Du, DZ., Li, L., Sun, X., Zhang, J. (eds) Algorithmic Aspects in Information and Management. AAIM 2019. Lecture Notes in Computer Science(), vol 11640. Springer, Cham. https://doi.org/10.1007/978-3-030-27195-4_4

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  • DOI: https://doi.org/10.1007/978-3-030-27195-4_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-27194-7

  • Online ISBN: 978-3-030-27195-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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