Approximate CVP_p in Time 2^{0.802 n}

Authors Friedrich Eisenbrand, Moritz Venzin

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Friedrich Eisenbrand
  • Ecole Polytechnique Fédérale de Lausanne, Switzerland
Moritz Venzin
  • Ecole Polytechnique Fédérale de Lausanne, Switzerland


The authors would like to thank the reviewers for their careful reviews and suggestions. The second author would like to thank Christoph Hunkenschröder, Noah Stephens-Davidowitz and Márton Naszódi for inspiring discussions.

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Friedrich Eisenbrand and Moritz Venzin. Approximate CVP_p in Time 2^{0.802 n}. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We show that a constant factor approximation of the shortest and closest lattice vector problem w.r.t. any 𝓁_p-norm can be computed in time 2^{(0.802 +ε) n}. This matches the currently fastest constant factor approximation algorithm for the shortest vector problem w.r.t. 𝓁₂. To obtain our result, we combine the latter algorithm w.r.t. 𝓁₂ with geometric insights related to coverings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Randomness, geometry and discrete structures
  • Shortest and closest vector problem
  • approximation algorithm
  • sieving
  • covering convex bodies


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  1. D. Aggarwal, D. Dadush, and N. Stephens-Davidowitz. Solving the closest vector problem in 2ⁿ time - the discrete gaussian strikes again! In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 563-582, October 2015. URL:
  2. Divesh Aggarwal, Huck Bennett, Alexander Golovnev, and Noah Stephens-Davidowitz. Fine-grained hardness of cvp (p) - everything that we can prove (and nothing else). arXiv preprint, 2019. URL:
  3. Divesh Aggarwal, Daniel Dadush, Oded Regev, and Noah Stephens-Davidowitz. Solving the shortest vector problem in 2n time using discrete gaussian sampling. In Proceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 733-742, 2015. Google Scholar
  4. Divesh Aggarwal and Priyanka Mukhopadhyay. Faster algorithms for SVP and CVP in the infinity norm. CoRR, abs/1801.02358, 2018. URL:
  5. Divesh Aggarwal and Noah Stephens-Davidowitz. (gap/s)eth hardness of svp. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, page 228–238, New York, NY, USA, 2018. Association for Computing Machinery. URL:
  6. Divesh Aggarwal and Noah Stephens-Davidowitz. Just take the average! An embarrassingly simple 2^n-time algorithm for SVP (and CVP). In 1st Symposium on Simplicity in Algorithms, SOSA 2018, January 7-10, 2018, New Orleans, LA, USA, pages 12:1-12:19, 2018. URL:
  7. Miklós Ajtai. The shortest vector problem in l2 is np-hard for randomized reductions (extended abstract). In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC ’98, page 10–19, New York, NY, USA, 1998. Association for Computing Machinery. URL:
  8. Miklós Ajtai, Ravi Kumar, and D. Sivakumar. A sieve algorithm for the shortest lattice vector problem. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 601-610, 2001. URL:
  9. Miklós Ajtai, Ravi Kumar, and D. Sivakumar. Sampling short lattice vectors and the closest lattice vector problem. In Proceedings of the 17th Annual IEEE Conference on Computational Complexity, Montréal, Québec, Canada, May 21-24, 2002, pages 53-57, 2002. URL:
  10. Sanjeev Arora. Probabilistic Checking of Proofs and Hardness of Approximation Problems. PhD thesis, University of California at Berkeley, Berkeley, CA, USA, 1995. UMI Order No. GAX95-30468. Google Scholar
  11. Shiri Artstein-Avidan and Boaz A Slomka. On weighted covering numbers and the levi-hadwiger conjecture. Israel Journal of Mathematics, 209(1):125-155, 2015. Google Scholar
  12. H. Bennett, A. Golovnev, and N. Stephens-Davidowitz. On the quantitative hardness of cvp. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 13-24, October 2017. URL:
  13. Ulrich Betke and Martin Henk. Intrinsic volumes and lattice points of crosspolytopes. Monatshefte für Mathematik, 115(1):27-33, 1993. URL:
  14. Johannes Blömer and Stefanie Naewe. Sampling methods for shortest vectors, closest vectors and successive minima. Theor. Comput. Sci., 410(18):1648-1665, 2009. URL:
  15. V. Chvatal. A greedy heuristic for the set-covering problem. Math. Oper. Res., 4(3):233–235, August 1979. URL:
  16. Daniel Dadush and Gábor Kun. Lattice sparsification and the approximate closest vector problem. Theory of Computing, 12(1):1-34, 2016. URL:
  17. Daniel Dadush, Chris Peikert, and Santosh S. Vempala. Enumerative lattice algorithms in any norm via m-ellipsoid coverings. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 580-589. IEEE Computer Society, 2011. URL:
  18. Irit Dinur, Guy Kindler, Ran Raz, and Shmuel Safra. Approximating CVP to within almost-polynomial factors is NP-hard. Combinatorica, 23(2):205-243, 2003. URL:
  19. Martin E. Dyer, Alan M. Frieze, and Ravi Kannan. A random polynomial time algorithm for approximating the volume of convex bodies. J. ACM, 38(1):1-17, 1991. URL:
  20. Friedrich Eisenbrand, Nicolai Hähnle, and Martin Niemeier. Covering cubes and the closest vector problem. In Proceedings of the 27th ACM Symposium on Computational Geometry, Paris, France, June 13-15, 2011, pages 417-423, 2011. URL:
  21. O. Goldreich, D. Micciancio, S. Safra, and Jean-Pierre Seifert. Approximating shortest lattice vectors is not harder than approximating closet lattice vectors. Inf. Process. Lett., 71(2):55–61, July 1999. URL:
  22. Peter Gruber. Convex and Discrete Geometry. Encyclopedia of Mathematics and its Applications. Springer, 2007. Google Scholar
  23. Ishay Haviv and Oded Regev. Tensor-based hardness of the shortest vector problem to within almost polynomial factors. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 469-477, 2007. Google Scholar
  24. Martin Henk, Jürgen Richter-Gebert, and Günter M Ziegler. Basic properties of convex polytopes. In Handbook of discrete and computational geometry, pages 243-270. CRC Press, 1997. Google Scholar
  25. Varun Jog and Venkat Anantharam. A geometric analysis of the awgn channel with a (σ, ρ)-power constraint. IEEE Transactions on Information Theory, April 2015. URL:
  26. Grigorii Anatol'evich Kabatiansky and Vladimir Iosifovich Levenshtein. On bounds for packings on a sphere and in space. Problemy Peredachi Informatsii, 14(1):3-25, 1978. Google Scholar
  27. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Math. Oper. Res., 12(3):415-440, 1987. URL:
  28. Subhash Khot. Hardness of approximating the shortest vector problem in lattices. J. ACM, 52(5):789–808, September 2005. URL:
  29. A. K. Lenstra, H. W. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515-534, 1982. URL:
  30. Hendrik W. Lenstra. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL:
  31. 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
  32. Daniele Micciancio. The shortest vector in a lattice is hard to approximate to within some constant. SIAM journal on Computing, 30(6):2008-2035, 2001. Google Scholar
  33. Daniele Micciancio and Panagiotis Voulgaris. A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 351-358, 2010. URL:
  34. Daniele Micciancio and Panagiotis Voulgaris. Faster exponential time algorithms for the shortest vector problem. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’10, page 1468–1480, USA, 2010. Society for Industrial and Applied Mathematics. Google Scholar
  35. Priyanka Mukhopadhyay. Faster provable sieving algorithms for the shortest vector problem and the closest vector problem on lattices in 𝓁_p norm. CoRR, abs/1907.04406, 2019. URL:
  36. Márton Naszódi and Moritz Venzin. Covering convex bodies and the closest vector problem. arXiv preprint, 2019. URL:
  37. Márton Naszódi. On some covering problems in geometry. Proceedings of the American Mathematical Society, 144, April 2014. URL:
  38. Xavier Pujol and Damien Stehlé. Solving the shortest lattice vector problem in time 2 2.465n. IACR Cryptology ePrint Archive, 2009:605, January 2009. Google Scholar
  39. Oded Regev. Lattices in computer science, lecture 8: 2^O(n) algorithm for svp, 2004. Google Scholar
  40. Rolf Schneider. Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2 edition, 2013. URL:
  41. Claus-Peter Schnorr. A hierarchy of polynomial time lattice basis reduction algorithms. Theoretical computer science, 53(2-3):201-224, 1987. Google Scholar
  42. P. van Emde Boas. Another NP-complete problem and the complexity of computing short vectors in a lattice. Technical Report 81-04, Mathematische Instituut, University of Amsterdam, 1981. Google Scholar
  43. Vijay V Vazirani. Approximation algorithms. Springer Science & Business Media, 2013. Google Scholar