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Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding

Authors Divesh Aggarwal, Yanlin Chen, Rajendra Kumar , Yixin Shen

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Lattices
  • Shortest Vector Problem
  • Discrete Gaussian Sampling
  • Time-Space Tradeoff
  • Quantum computation
  • Bounded distance decoding

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Abstract

The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results.  
1) A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer 4 ≤ q ≤ √n, our algorithm takes q^{13n+o(n)} time and requires poly(n)⋅ q^{16n/q²} memory. This tradeoff which ranges from enumeration (q = √n) to sieving (q constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. 
2) A quantum algorithm that runs in time 2^{0.9533n+o(n)} and requires 2^{0.5n+o(n)} classical memory and poly(n) qubits. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [Divesh Aggarwal et al., 2015] that has a time and space complexity 2^{n+o(n)}. 
3) A classical algorithm for SVP that runs in time 2^{1.741n+o(n)} time and 2^{0.5n+o(n)} space. This improves over an algorithm of [Yanlin Chen et al., 2018] that has the same space complexity. 
The time complexity of our classical and quantum algorithms are expressed using a quantity related to the kissing number of a lattice. A known upper bound of this quantity is 2^{0.402n}, but in practice for most lattices, it can be much smaller and even 2^o(n). In that case, our classical algorithm runs in time 2^{1.292n} and our quantum algorithm runs in time 2^{0.750n}.

Cite As Get BibTex

Divesh Aggarwal, Yanlin Chen, Rajendra Kumar, and Yixin Shen. Improved (Provable) Algorithms for the Shortest Vector Problem via Bounded Distance Decoding. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.STACS.2021.4

Author Details

Divesh Aggarwal
  • Centre for Quantum Technologies, Singapore
  • National University of Singapore, Singapore
Yanlin Chen
  • Institute of Information Science, Academia Sinica, Taipei, Taiwan
Rajendra Kumar
  • IIT Kanpur, India
  • National University of Singapore, Singapore
Yixin Shen
  • Université de Paris, IRIF, CNRS, F-75006, France

Funding

  • Aggarwal, Divesh: This work was partially supported by the Singapore Ministry of Education under grant MOE2019-T2-1-145, National Research Foundation under grant R-710-000-012-135.
  • Kumar, Rajendra: Supported in part by National Research Foundation Singapore under its AI Singapore Programme [Award Number: AISG-RP-2018-005].

Acknowledgements

We would like to thank Pierre-Alain Fouque, Paul Kirchner, Amaury Pouly and Noah Stephens-Davidowitz for useful comments and suggestions.

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