Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm

Authors Divesh Aggarwal, Priyanka Mukhopadhyay



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2018.35.pdf
  • Filesize: 0.5 MB
  • 13 pages

Document Identifiers

Author Details

Divesh Aggarwal
  • Centre for Quantum Technologies and School of Computing, National University of Singapore
Priyanka Mukhopadhyay
  • Centre for Quantum Technologies, National University of Singapore

Cite AsGet BibTex

Divesh Aggarwal and Priyanka Mukhopadhyay. Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ISAAC.2018.35

Abstract

Ajtai, Kumar and Sivakumar [Ajtai et al., 2001] gave the first 2^O(n) algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. The algorithm starts with N in 2^O(n) randomly chosen vectors in the lattice and employs a sieving procedure to iteratively obtain shorter vectors in the lattice, and eventually obtaining the shortest non-zero vector. The running time of the sieving procedure is quadratic in N. Subsequent works [Arvind and Joglekar, 2008; Blömer and Naewe, 2009] generalized the algorithm to other norms. We study this problem for the special but important case of the l_infty norm. We give a new sieving procedure that runs in time linear in N, thereby improving the running time of the algorithm for SVP in the l_infty norm. As in [Ajtai et al., 2002; Blömer and Naewe, 2009], we also extend this algorithm to obtain significantly faster algorithms for approximate versions of the shortest vector problem and the closest vector problem (CVP) in the l_infty norm. We also show that the heuristic sieving algorithms of Nguyen and Vidick [Nguyen and Vidick, 2008] and Wang et al. [Wang et al., 2011] can also be analyzed in the l_infty norm. The main technical contribution in this part is to calculate the expected volume of intersection of a unit ball centred at origin and another ball of a different radius centred at a uniformly random point on the boundary of the unit ball. This might be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Lattice
  • Shortest Vector Problem
  • Closest Vector Problem
  • l_infty norm

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Divesh Aggarwal, Daniel Dadush, Oded Regev, and Noah Stephens-Davidowitz. Solving the Shortest Vector Problem in 2ⁿ time via Discrete Gaussian Sampling. In STOC, 2015. Full version available at URL: https://arxiv.org/abs/1412.7994.
  2. Divesh Aggarwal, Daniel Dadush, and Noah Stephens-Davidowitz. Solving the Closest Vector Problem in 2^n Time-The Discrete Gaussian Strikes Again! In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 563-582. IEEE, 2015. Google Scholar
  3. Divesh Aggarwal and Priyanka Mukhopadhyay. Faster algorithms for SVP and CVP in the 𝓁_∞ norm. arXiv preprint, 2018. URL: http://arxiv.org/abs/1801.02358.
  4. Divesh Aggarwal and Noah Stephens-Davidowitz. Just Take the Average! An Embarrassingly Simple 2ⁿ-Time Algorithm for SVP (and CVP). arXiv preprint, 2017. URL: http://arxiv.org/abs/1709.01535.
  5. Miklós Ajtai, Ravi Kumar, and D. Sivakumar. A Sieve Algorithm for the Shortest Lattice Vector Problem. In STOC, pages 601-610, 2001. URL: http://dx.doi.org/10.1145/380752.380857.
  6. Miklós Ajtai, Ravi Kumar, and D. Sivakumar. Sampling short lattice vectors and the closest lattice vector problem. In CCC, pages 41-45, 2002. Google Scholar
  7. Vikraman Arvind and Pushkar S Joglekar. Some sieving algorithms for lattice problems. In LIPIcs-Leibniz International Proceedings in Informatics, volume 2. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2008. Google Scholar
  8. L. Babai. On Lovász' lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1-13, 1986. URL: http://dx.doi.org/10.1007/BF02579403.
  9. Anja Becker, Léo Ducas, Nicolas Gama, and Thijs Laarhoven. New directions in nearest neighbor searching with applications to lattice sieving. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 10-24. Society for Industrial and Applied Mathematics, 2016. Google Scholar
  10. Huck Bennett, Alexander Golovnev, and Noah Stephens-Davidowitz. On the Quantitative Hardness of CVP. arXiv preprint, 2017. URL: http://arxiv.org/abs/1704.03928.
  11. Johannes Blömer and Stefanie Naewe. Sampling methods for shortest vectors, closest vectors and successive minima. Theoretical Computer Science, 410(18):1648-1665, 2009. Google Scholar
  12. Matthijs J Coster, Antoine Joux, Brian A LaMacchia, Andrew M Odlyzko, Claus-Peter Schnorr, and Jacques Stern. Improved low-density subset sum algorithms. computational complexity, 2(2):111-128, 1992. Google Scholar
  13. Léo Ducas, Tancrède Lepoint, Vadim Lyubashevsky, Peter Schwabe, Gregor Seiler, and Damien Stehlé. CRYSTALS-Dilithium: Digital Signatures from Module Lattices. Technical report, IACR Cryptology ePrint Archive, 2017: 633, 2017. Google Scholar
  14. Friedrich Eisenbrand, Nicolai Hähnle, and Martin Niemeier. Covering cubes and the closest vector problem. In Proceedings of the twenty-seventh annual symposium on Computational geometry, pages 417-423. ACM, 2011. Google Scholar
  15. O. Goldreich, D. Micciancio, S. Safra, and J.-P. Seifert. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Information Processing Letters, 71(2):55-61, 1999. URL: http://dx.doi.org/10.1016/S0020-0190(99)00083-6.
  16. Ravi Kannan. Minkowski’s Convex Body Theorem and Integer Programming. Mathematics of Operations Research, 12(3):pp. 415-440, 1987. URL: http://www.jstor.org/stable/3689974.
  17. Susan Landau and Gary Lee Miller. Solvability by radicals is in polynomial time. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 140-151. ACM, 1983. Google Scholar
  18. A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261(4):515-534, 1982. URL: http://dx.doi.org/10.1007/BF01457454.
  19. Hendrik W Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of operations research, 8(4):538-548, 1983. Google Scholar
  20. Mingjie Liu, Xiaoyun Wang, Guangwu Xu, and Xuexin Zheng. Shortest lattice vectors in the presence of gaps. IACR Cryptology ePrint Archive, 2011:139, 2011. Google Scholar
  21. Daniele Micciancio and Panagiotis Voulgaris. A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM Journal on Computing, 42(3):1364-1391, 2013. Google Scholar
  22. Phong Q Nguyen and Jacques Stern. The two faces of lattices in cryptology. In Cryptography and lattices, pages 146-180. Springer, 2001. Google Scholar
  23. Phong Q Nguyen and Thomas Vidick. Sieve algorithms for the shortest vector problem are practical. Journal of Mathematical Cryptology, 2(2):181-207, 2008. Google Scholar
  24. Chris Peikert et al. A decade of lattice cryptography. Foundations and Trendsregistered in Theoretical Computer Science, 10(4):283-424, 2016. Google Scholar
  25. Xavier Pujol and Damien Stehlé. Solving the Shortest Lattice Vector Problem in Time 2^2.465n. IACR Cryptology ePrint Archive, 2009:605, 2009. Google Scholar
  26. Oded Regev. Lecture notes on lattices in computer science, 2009. Google Scholar
  27. Xiaoyun Wang, Mingjie Liu, Chengliang Tian, and Jingguo Bi. Improved Nguyen-Vidick heuristic sieve algorithm for shortest vector problem. In Proceedings of the 6th ACM Symposium on Information, Computer and Communications Security, pages 1-9. ACM, 2011. Google Scholar
  28. Wei Wei, Mingjie Liu, and Xiaoyun Wang. Finding Shortest Lattice Vectors in the Presence of Gaps. In Topics in Cryptology - CT-RSA 2015, The Cryptographer’s Track at the RSA Conference 2015, San Francisco, CA, USA, April 20-24, 2015. Proceedings, pages 239-257, 2015. URL: http://dx.doi.org/10.1007/978-3-319-16715-2_13.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail