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International Conference on Machine Learning for Cyber Security

ML4CS 2019: Machine Learning for Cyber Security pp 154–172Cite as

An Enumeration-Like Vector Sampling Method for Solving Approximate SVP

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11806))

Abstract

Lattice reduction with random sampling is a kind of randomized heuristic algorithm for solving approximate Shortest Vector Problem (SVP). In this paper, we propose a lattice vector sampling method for solving approximate SVP. Firstly, we apply enumeration techniques into vector sampling using natural number’s representation (NNR), enlightened by discrete pruning. Secondly, to find optimal parameters for the enumeration-like sampling method, we study the statistical properties of a structured candidate vector set, and give a parameter calculation strategy for minimizing the sampling time. This new sampling method is a universal framework that can be embedded into most of the sampling-reduction algorithms. The experimental result shows that sampling reduction algorithm with the new sampling method embedded runs faster than the original Restricted Reduction (RR) algorithm within 90 dimensions.

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References

  1. http://www.latticechallenge.org/svp-challenge/

  2. http://www.shoup.net/ntl/

  3. Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 601–610. ACM (2001)

    Google Scholar 

  4. Aono, Y., Nguyen, P.Q.: Random sampling revisited: lattice enumeration with discrete pruning. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10211, pp. 65–102. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56614-6_3

    Chapter  Google Scholar 

  5. Becker, A., Ducas, L., Gama, N., Laarhoven, T.: New directions in nearest neighbor searching with applications to lattice sieving. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 10–24. Society for Industrial and Applied Mathematics (2016)

    Google Scholar 

  6. Boneh, D., et al.: Twenty years of attacks on the RSA cryptosystem. Not.-Am. Math. Soc. 46, 203–213 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Buchmann, J., Ludwig, C.: Practical lattice basis sampling reduction. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 222–237. Springer, Heidelberg (2006). https://doi.org/10.1007/11792086_17

    Chapter  Google Scholar 

  8. Coppersmith, D.: Finding small solutions to small degree polynomials. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 20–31. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44670-2_3

    Chapter  Google Scholar 

  9. van Emde Boas, P.: Another NP-complete problem and the complexity of computing short vectors in a lattice. Tecnical report, Department of Mathmatics, University of Amsterdam (1981)

    Google Scholar 

  10. Fukase, M., Kashiwabara, K.: An accelerated algorithm for solving SVP based on statistical analysis. J. Inf. Process. 23(1), 67–80 (2015)

    Google Scholar 

  11. 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 

  12. Kannan, R.: Improved algorithms for integer programming and related lattice problems. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pp. 193–206. ACM (1983)

    Google Scholar 

  13. Khot, S.: Hardness of approximating the shortest vector problem in lattices. J. ACM (JACM) 52(5), 789–808 (2005)

    Article  MathSciNet  Google Scholar 

  14. 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  MATH  Google Scholar 

  15. Micciancio, D.: Lattice-based cryptography. In: van Tilborg, H.C.A., Jajodia, S. (eds.) Encyclopedia of Cryptography and Security, pp. 713–715. Springer, Boston (2011). https://doi.org/10.1007/978-1-4419-5906-5_417

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM (JACM) 56(6), 34 (2009)

    Article  MathSciNet  Google Scholar 

  18. Schnorr, C.P.: Lattice reduction by random sampling and birthday methods. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 145–156. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36494-3_14

    Chapter  Google Scholar 

  19. Teruya, T., Kashiwabara, K., Hanaoka, G.: Fast lattice basis reduction suitable for massive parallelization and its application to the shortest vector problem. In: Abdalla, M., Dahab, R. (eds.) PKC 2018. LNCS, vol. 10769, pp. 437–460. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76578-5_15

    Chapter  Google Scholar 

  20. Van Hoeij, M.: Factoring polynomials and the Knapsack problem. J. Number Theory 95(2), 167–189 (2002)

    Article  MathSciNet  Google Scholar 

  21. Yasuda, M., Yokoyama, K., Shimoyama, T., Kogure, J., Koshiba, T.: Analysis of decreasing squared-sum of gram-schmidt lengths for short lattice vectors. J. Math. Cryptol. 11(1), 1–24 (2017)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Henan Key Laboratory of Network Cryptography Technology, Zhengzhou, China

    Luan Luan, Chunxiang Gu & Yonghui Zheng

Authors
  1. Luan Luan
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  2. Chunxiang Gu
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  3. Yonghui Zheng
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Corresponding author

Correspondence to Luan Luan .

Editor information

Editors and Affiliations

  1. Xidian University, Xi’an, China

    Xiaofeng Chen

  2. Fujian Normal University, Fuzhou, China

    Xinyi Huang

  3. Swinburne University of Technology, Hawthorn, VIC, Australia

    Jun Zhang

Appendix A The Pseudocode of Algorithms

Appendix A The Pseudocode of Algorithms

figure a
figure b
figure c

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Luan, L., Gu, C., Zheng, Y. (2019). An Enumeration-Like Vector Sampling Method for Solving Approximate SVP. In: Chen, X., Huang, X., Zhang, J. (eds) Machine Learning for Cyber Security. ML4CS 2019. Lecture Notes in Computer Science(), vol 11806. Springer, Cham. https://doi.org/10.1007/978-3-030-30619-9_12

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  • DOI: https://doi.org/10.1007/978-3-030-30619-9_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-30618-2

  • Online ISBN: 978-3-030-30619-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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