Skip to main content
SpringerLink
Log in

BS: Blockwise Sieve Algorithm for Finding Short Vectors from Sublattices

  • Conference paper
  • First Online:
Information and Communications Security (ICICS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13407))

Included in the following conference series:

Abstract

The Shortest Vector Problem is a crucial part of the lattice theory and a central lattice problem in analyzing lattice-based cryptography. This work provides a new algorithm that finds a short vector by calling the sieve oracle in projected sublattices orthogonal to each other. We first propose the Block Sieve algorithm. With blockwise sieving, proper insertion and reduction, the coordinates of the right end of vector v can be recovered. The algorithm moves the block to recover the other coordinates. We continue to optimize the algorithm and propose the Progressive Block Sieve algorithm, employing techniques such as skipping to accelerate the procedure. In a d-dimensional lattice, smaller sieve calls in (\(d-\varTheta ({d}/\ln {d})\))-dimensional sublattices are enough to find a short vector. We compare the experimental results on different lattices to test the performance of the new approach. On challenge lattices, our algorithm takes less time and fewer tours than original reduction algorithms to reach a similar outcome. As an application of the new algorithm, we test the performance of solving Learning With Errors problem. Our algorithm is able to solve the instances about 5% faster than sieving.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aggarwal, D., Li, J., Nguyen, P.Q., Stephens-Davidowitz, N.: Slide reduction, revisited—filling the gaps in SVP approximation. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12171, pp. 274–295. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56880-1_10

    Chapter  MATH  Google Scholar 

  2. Ajtai, M.: Generating hard instances of the short basis problem. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 1–9. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48523-6_1

    Chapter  Google Scholar 

  3. Albrecht, M.R., Bai, S., Fouque, P.-A., Kirchner, P., Stehlé, D., Wen, W.: Faster enumeration-based lattice reduction: root Hermite factor \(k^{1/(2k)}\) time \(k^{k/8+o(k)}\). In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12171, pp. 186–212. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56880-1_7

    Chapter  Google Scholar 

  4. Albrecht, M.R., Bai, S., Li, J., Rowell, J.: Lattice reduction with approximate enumeration oracles: practical algorithms and concrete performance. Cryptology ePrint Archive, Report 2020/1260 (2020). https://eprint.iacr.org/2020/1260

  5. Albrecht, M.R., Ducas, L., Herold, G., Kirshanova, E., Postlethwaite, E.W., Stevens, M.: The general sieve kernel and new records in lattice reduction. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11477, pp. 717–746. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17656-3_25

    Chapter  MATH  Google Scholar 

  6. Albrecht, M.R., Göpfert, F., Virdia, F., Wunderer, T.: Revisiting the expected cost of solving uSVP and applications to LWE. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 297–322. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_11

    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. 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, SODA 2016, pp. 10–24. Society for Industrial and Applied Mathematics (2016)

    Google Scholar 

  9. Chen, Y.: Réduction de réseau et sécurité concrète du chiffrement complètement homomorphe. Ph.D. thesis, Higher Normal School - PSL (2013). https://www.theses.fr/2013PA077242, thèse de doctorat dirigée par Nguyen, Phong-Quang Informatique Paris 7 2013

  10. 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 

  11. Ducas, L.: Shortest vector from lattice sieving: a few dimensions for free. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 125–145. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_5

    Chapter  Google Scholar 

  12. Gama, N., Nguyen, P.Q.: Finding short lattice vectors within Mordell’s inequality. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC 2008, pp. 207–216. Association for Computing Machinery, New York (2008). https://doi.org/10.1145/1374376.1374408

  13. Gama, N., Nguyen, P.Q.: Predicting lattice reduction. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 31–51. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78967-3_3

    Chapter  Google Scholar 

  14. 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 

  15. Hanrot, G., Pujol, X., Stehlé, D.: Analyzing blockwise lattice algorithms using dynamical systems. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 447–464. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_25

    Chapter  Google Scholar 

  16. Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054868

    Chapter  Google Scholar 

  17. 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 

  18. Lenstra, A.K., Lenstra, H.W., Lovasz, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982). https://doi.org/10.1007/BF01457454

  19. Lindner, R., Peikert, C.: Better key sizes (and attacks) for LWE-based encryption. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 319–339. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19074-2_21

    Chapter  Google Scholar 

  20. Micciancio, D., Walter, M.: Fast lattice point enumeration with minimal overhead. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, pp. 276–294. Society for Industrial and Applied Mathematics (2015)

    Google Scholar 

  21. 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 

  22. Nguyen, P., Valle, B. (eds.): The LLL Algorithm. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-02295-1

    Book  MATH  Google Scholar 

  23. Pohst, M.: On the computation of lattice vectors of minimal length, successive minima and reduced bases with applications. SIGSAM Bull. 15(1), 37–44 (1981). https://doi.org/10.1145/1089242.1089247

    Article  MATH  Google Scholar 

  24. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6) (2009). https://doi.org/10.1145/1568318.1568324

  25. Schneider, M., Gama, N.: SVP challenge. [EB/OL]. https://www.latticechallenge.org/svp-challenge. Accessed 25 June 2021

  26. 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 

  27. 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 

  28. T. F. Development Team: FPLLL, a lattice reduction library, Version: 5.4.1 (2021). https://github.com/fplll/fplll

  29. T. F. Development Team: FPYLLL, a Python wraper for the FPLLL lattice reduction library, Version: 0.5.6 (2021). https://github.com/fplll/fpylll

  30. T. F. Development Team: The general sieve kernel (g6k) (2021). https://github.com/fplll/fpylll

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61872449, 62125205).

Author information

Authors and Affiliations

  1. Strategic Support Force Information Engineering University, Zhengzhou, 450001, China

    Jinzheng Cao & Qingfeng Cheng

  2. School of Cyber Engineering, Xidian University, Xi’an, 710071, China

    Xinghua Li

  3. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yanbin Pan

Authors
  1. Jinzheng Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingfeng Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xinghua Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yanbin Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingfeng Cheng .

Editor information

Editors and Affiliations

  1. University of Malaga, Malaga, Spain

    Cristina Alcaraz

  2. University of Surrey, Guildford, UK

    Liqun Chen

  3. University of Kent, Canterbury, UK

    Shujun Li

  4. University of Milan, Milan, Italy

    Pierangela Samarati

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cao, J., Cheng, Q., Li, X., Pan, Y. (2022). BS: Blockwise Sieve Algorithm for Finding Short Vectors from Sublattices. In: Alcaraz, C., Chen, L., Li, S., Samarati, P. (eds) Information and Communications Security. ICICS 2022. Lecture Notes in Computer Science, vol 13407. Springer, Cham. https://doi.org/10.1007/978-3-031-15777-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-15777-6_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-15776-9

  • Online ISBN: 978-3-031-15777-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation