Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One

Author Noah Stephens-Davidowitz

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Noah Stephens-Davidowitz

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Noah Stephens-Davidowitz. Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any gamma <= 1 + O(log n/n), we obtain an efficient dimension-preserving reduction from gamma^{O(n/log n)}-SVP to gamma-GapSVP and an efficient dimension-preserving reduction from gamma^{O(n)}-CVP to gamma-GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when gamma = 1. For SVP, we actually obtain something slightly stronger than a search-to-decision reduction - we reduce gamma^{O(n/log n)}-SVP to gamma-unique SVP, a potentially easier problem than gamma-GapSVP.
  • Lattices
  • SVP
  • CVP


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  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. Google Scholar
  2. Divesh Aggarwal and Chandan Dubey. Improved hardness results for unique Shortest Vector Problem, 2015. URL:
  3. Dorit Aharonov and Oded Regev. Lattice problems in NP intersect coNP. Journal of the ACM, 52(5):749-765, 2005. Preliminary version in FOCS'04. Google Scholar
  4. Miklós Ajtai. Generating hard instances of lattice problems. In STOC, pages 99-108. ACM, 1996. Google Scholar
  5. Miklós Ajtai. The Shortest Vector Problem in L2 is NP-hard for randomized reductions. In STOC, 1998. URL:
  6. Miklós Ajtai, Ravi Kumar, and D. Sivakumar. A sieve algorithm for the shortest lattice vector problem. In STOC, pages 601-610, 2001. Google Scholar
  7. L. Babai. On Lovász' lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1-13, 1986. URL:
  8. Shi Bai, Weiqiang Wen, and Damien Stehlé. Improved reduction from the Bounded Distance Decoding problem to the unique Shortest Vector Problem in lattices. In ICALP, 2016. Google Scholar
  9. W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296(4):625-635, 1993. URL:
  10. U. Betke, M. Henk, and J.M. Wills. Successive-minima-type inequalities. Discrete &Computational Geometry, 9(1):165-175, 1993. URL:
  11. Zvika Brakerski and Vinod Vaikuntanathan. Efficient fully homomorphic encryption from (standard) LWE. In FOCS, pages 97-106. IEEE, 2011. Google Scholar
  12. Zvika Brakerski and Vinod Vaikuntanathan. Lattice-based FHE as secure as PKE. In ITCS, pages 1-12, 2014. Google Scholar
  13. Jin-Yi Cai and Ajay Nerurkar. Approximating the SVP to within a factor (1+1/dim^ε) is NP-hard under randomized reductions. Journal of Computer and System Sciences, 59(2):221-239, 1999. URL:
  14. Daniel Dadush and Gabor Kun. Lattice sparsification and the approximate Closest Vector Problem. In SODA, 2013. Google Scholar
  15. Daniel Dadush, Chris Peikert, and Santosh Vempala. Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In FOCS, pages 580-589. IEEE, 2011. Google Scholar
  16. Daniel Dadush, Oded Regev, and Noah Stephens-Davidowitz. On the Closest Vector Problem with a distance guarantee. In CCC, pages 98-109, 2014. URL:
  17. Craig Gentry. Fully homomorphic encryption using ideal lattices. In STOC, pages 169-178. ACM, New York, 2009. Google Scholar
  18. Craig Gentry, Chris Peikert, and Vinod Vaikuntanathan. Trapdoors for hard lattices and new cryptographic constructions. In STOC, pages 197-206, 2008. Google Scholar
  19. Oded Goldreich, Daniele Micciancio, Shmuel Safra, and Jean-Paul Seifert. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Information Processing Letters, 71(2):55-61, 1999. URL:
  20. Ishay Haviv and Oded Regev. Tensor-based hardness of the Shortest Vector Problem to within almost polynomial factors. Theory of Computing, 8(23):513-531, 2012. Preliminary version in STOC'07. Google Scholar
  21. Antoine Joux and Jacques Stern. Lattice reduction: A toolbox for the cryptanalyst. Journal of Cryptology, 11(3):161-185, 1998. Google Scholar
  22. G. A. Kabatjanskiĭ and V. I. Levenšteĭn. Bounds for packings on the sphere and in space. Problemy Peredači Informacii, 14(1):3-25, 1978. Google Scholar
  23. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12(3):pp. 415-440, 1987. URL:
  24. Subhash Khot. Hardness of approximating the Shortest Vector Problem in lattices. Journal of the ACM, 52(5):789-808, September 2005. Preliminary version in FOCS'04. Google Scholar
  25. S. Ravi Kumar and D. Sivakumar. On the unique shortest lattice vector problem. Theoretical Computer Science, 255(1–2):641-648, 2001. URL:
  26. A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261(4):515-534, 1982. URL:
  27. Hendrik W Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of operations research, 8(4):538-548, 1983. Google Scholar
  28. Yi-Kai Liu, Vadim Lyubashevsky, and Daniele Micciancio. On Bounded Distance Decoding for general lattices. In RANDOM, 2006. Google Scholar
  29. Vadim Lyubashevsky and Daniele Micciancio. On Bounded Distance Decoding, unique shortest vectors, and the minimum distance problem. In Advances in Cryptology - CRYPTO 2009, volume 5677 of LNCS, pages 577-594. Springer, 2009. URL:
  30. Vadim Lyubashevsky, Chris Peikert, and Oded Regev. On ideal lattices and Learning with Errors over rings. In EUROCRYPT, 2010. Google Scholar
  31. Daniele Micciancio. The Shortest Vector Problem is NP-hard to approximate to within some constant. SIAM Journal on Computing, 30(6):2008-2035, March 2001. Preliminary version in FOCS 1998. Google Scholar
  32. Daniele Micciancio. Efficient reductions among lattice problems. In SODA, pages 84-93. ACM, New York, 2008. Google Scholar
  33. Daniele Micciancio and Shafi Goldwasser. Complexity of Lattice Problems: a cryptographic perspective, volume 671 of The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston, Massachusetts, March 2002. Google Scholar
  34. Daniele Micciancio and Oded Regev. Worst-case to average-case reductions based on Gaussian measures. SIAM Journal on Computing, 37(1):267-302, 2007. Google Scholar
  35. Phong Q Nguyen and Jacques Stern. The two faces of lattices in cryptology. In Cryptography and lattices, pages 146-180. Springer, 2001. Google Scholar
  36. Andrew M Odlyzko. The rise and fall of knapsack cryptosystems. Cryptology and computational number theory, 42:75-88, 1990. Google Scholar
  37. Chris Peikert. Public-key cryptosystems from the worst-case Shortest Vector Problem. In STOC, pages 333-342. ACM, 2009. Google Scholar
  38. Chris Peikert and Alon Rosen. Lattices that admit logarithmic worst-case to average-case connection factors. In STOC, 2007. Google Scholar
  39. Xavier Pujol and Damien Stehlé. Solving the shortest lattice vector problem in time 2^2.465 n. IACR Cryptology ePrint Archive, 2009:605, 2009. Google Scholar
  40. Oded Regev. On lattices, Learning with Errors, random linear codes, and cryptography. Journal of the ACM, 56(6):Art. 34, 40, 2009. URL:
  41. Noah Stephens-Davidowitz. Dimension-preserving reductions between lattice problems, 2015. URL:
  42. Noah Stephens-Davidowitz. Discrete Gaussian sampling reduces to CVP and SVP. In SODA, 2016. Google Scholar
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