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A New Lattice Sieving Algorithm Base on Angular Locality-Sensitive Hashing

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Information Security and Cryptology (Inscrypt 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10726))

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Abstract

Currently, the space requirement of sieving algorithms to solve the shortest vector problem (SVP) grows as \(2^{0.2075n+o(n)}\), where n is the lattice dimension. In high dimensions, the memory requirement makes them uncompetitive with enumeration algorithms. Shi Bai et al. presents a filtered triple sieving algorithm that breaks the bottleneck with memory \( 2^{0.1887n+o(n)}\) and time \( 2^{0.481n+o(n)}\).

Benefiting from the angular locality-sensitive hashing (LSH) method, our proposed algorithm runs in time \(2^{0.4098n+o(n)}\) with the same space complexity \(2^{0.1887n+o(n)}\) as the filtered triple sieving algorithm. Our experiment demonstrates that the proposed algorithm achieves the desired results. Furthermore, we use the proposed algorithm to solve the closest vector problem (CVP) with the lowest space complexity as far as we know in the literature.

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Notes

  1. 1.

    The average probability that a distant (non-reducing) vector \(\varvec{w}\) collides with \(\varvec{v}\) in at least one of the t hash tables [20].

References

  1. Lenstra, H.W., Lenstra, A.K., Lovfiasz, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Kannan, R.: Improved algorithms for integer programming and related lattice problems. In: ACM Symposium on Theory of Computing, 25–27 April 1983, Boston, Massachusetts, USA, pp. 193–206 (1983)

    Google Scholar 

  3. Schnorr, C.P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. In: Budach, L. (ed.) FCT 1991. LNCS, vol. 529, pp. 68–85. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54458-5_51

    Chapter  Google Scholar 

  4. Gama, N., Nguyen, P.Q., Regev, O.: Lattice enumeration using extreme pruning. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 257–278. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_13

    Chapter  Google Scholar 

  5. Chen, Y., Nguyen, P.Q.: BKZ 2.0: better lattice security estimates. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 1–20. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_1

    Chapter  Google Scholar 

  6. Micciancio, D., Walter, M.: Practical, predictable lattice basis reduction. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9665, pp. 820–849. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49890-3_31

    Chapter  Google Scholar 

  7. Aono, Y., Wang, Y., Hayashi, T., Takagi, T.: Improved progressive BKZ algorithms and their precise cost estimation by sharp simulator. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9665, pp. 789–819. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49890-3_30

    Chapter  Google Scholar 

  8. Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: ACM Symposium on Theory of Computing, pp. 601–610 (2002)

    Google Scholar 

  9. Nguyen, P.Q., Vidick, T.: Sieve algorithms for the shortest vector problem are practical. J. Math. Cryptology 2(2), 181–207 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, X., Liu, M., Tian, C., Bi, J.: Improved Nguyen-Vidick heuristic sieve algorithm for shortest vector problem. In: ACM Symposium on Information, Computer and Communications Security, ASIACCS 2011, Hong Kong, China, March 2011, pp. 1–9 (2011)

    Google Scholar 

  11. Micciancio, D., Voulgaris, P.: Faster exponential time algorithms for the shortest vector problem. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 1468–1480 (2010)

    Google Scholar 

  12. Pujol, X., Stehl, D.: Solving the shortest lattice vector problem in time 2 2.465n. IACR Cryptology ePrint Archive, vol. 2009 (2006)

    Google Scholar 

  13. Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. In: ACM Symposium on Theory of Computing, pp. 351–358 (2010)

    Google Scholar 

  14. Aggarwal, D., Dadush, D., Regev, O., Stephens-Davidowitz, N.: Solving the shortest vector problem in 2 n time using discrete Gaussian sampling: extended abstract. In: Forty-Seventh ACM Symposium on Theory of Computing, pp. 733–742 (2015)

    Google Scholar 

  15. Charikar, M.S.: Similarity estimation techniques from rounding algorithms. In: Thiry-Fourth ACM Symposium on Theory of Computing, pp. 380–388 (2002)

    Google Scholar 

  16. Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. In: Theory of Computing, no. 11, pp. 604–613 (2000)

    Google Scholar 

  17. Becker, A., Laarhoven, T.: Efficient (ideal) lattice sieving using cross-polytope LSH. In: Pointcheval, D., Nitaj, A., Rachidi, T. (eds.) AFRICACRYPT 2016. LNCS, vol. 9646, pp. 3–23. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-31517-1_1

    Chapter  Google Scholar 

  18. Becker, A., Ducas, L., Gama, N., Laarhoven, T.: New directions in nearest neighbor searching with applications to lattice sieving. In: Twenty-Seventh ACM-SIAM Symposium on Discrete Algorithms, pp. 10–24 (2016)

    Google Scholar 

  19. Shi, B.: Tuple lattice sieving. LMS J. Comput. Math. 19(A), 146–162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Laarhoven, T.: Sieving for shortest vectors in lattices using angular locality-sensitive hashing. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 3–22. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_1

    Chapter  Google Scholar 

  21. Panigrahy, R.: Entropy based nearest neighbor search in high dimensions. In: SODA 2006: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1186–1195 (2005)

    Google Scholar 

  22. Goldstein, D., Mayer, A.: On the equidistribution of hecke points. Forum Mathematicum 15(2), 165–189 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Goldstein, D.M.A.: SVP challenge (2010). http://www.latticechallenge.org

  24. Schneider, M.: Sieving for shortest vectors in ideal lattices. In: Youssef, A., Nitaj, A., Hassanien, A.E. (eds.) AFRICACRYPT 2013. LNCS, vol. 7918, pp. 375–391. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38553-7_22

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. College of Information Engineering, Shenzhen University, Shenzhen, 518060, China

    Ping Wang

  2. College of Computer Science and Software, Shenzhen University, Shenzhen, 518060, China

    Dongdong Shang

Authors
  1. Ping Wang
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  2. Dongdong Shang
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Correspondence to Ping Wang .

Editor information

Editors and Affiliations

  1. Xidian University, Xi’an, China

    Xiaofeng Chen

  2. SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

  3. Columbia University, New York, New York, USA

    Moti Yung

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Wang, P., Shang, D. (2018). A New Lattice Sieving Algorithm Base on Angular Locality-Sensitive Hashing. In: Chen, X., Lin, D., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2017. Lecture Notes in Computer Science(), vol 10726. Springer, Cham. https://doi.org/10.1007/978-3-319-75160-3_6

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  • DOI: https://doi.org/10.1007/978-3-319-75160-3_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-75159-7

  • Online ISBN: 978-3-319-75160-3

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