Skip to main content
Book cover

IACR International Conference on Public-Key Cryptography

PKC 2021: Public-Key Cryptography – PKC 2021 pp 184–214Cite as

On the Integer Polynomial Learning with Errors Problem

  • 650 Accesses

Part of the Lecture Notes in Computer Science book series (LNSC,volume 12710)

Abstract

Several recent proposals of efficient public-key encryption are based on variants of the polynomial learning with errors problem (PLWE\(^f\)) in which the underlying polynomial ring \(\mathbb {Z}_q[x]/f\) is replaced with the (related) modular integer ring \(\mathbb {Z}_{f(q)}\); the corresponding problem is known as Integer Polynomial Learning with Errors (I-PLWE\(^f\)). Cryptosystems based on I-PLWE\(^f\) and its variants can exploit optimised big-integer arithmetic to achieve good practical performance, as exhibited by the ThreeBears cryptosystem. Unfortunately, the average-case hardness of I-PLWE\(^f\) and its relation to more established lattice problems have to date remained unclear.

We describe the first polynomial-time average-case reductions for the search variant of I-PLWE\(^f\), proving its computational equivalence with the search variant of its counterpart problem PLWE\(^f\). Our reductions apply to a large class of defining polynomials f. To obtain our results, we employ a careful adaptation of Rényi divergence analysis techniques to bound the impact of the integer ring arithmetic carries on the error distributions. As an application, we present a deterministic public-key cryptosystem over integer rings. Our cryptosystem, which resembles ThreeBears, enjoys one-way (OW-CPA) security provably based on the search variant of I-PLWE\(^f\).

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-75245-3_8
  • Chapter length: 31 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   109.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-75245-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   149.99
Price excludes VAT (USA)
Learn about institutional subscriptions

Notes

  1. 1.

    As commonly done, we also impose irreducibility of f in the problem definitions, to avoid weaknesses such as those pointed out in [BCF20].

References

  1. Alkim, E., et al.: Frodo: Candidate to NIS (2017)

    Google Scholar 

  2. Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast cryptographic primitives and circular-secure encryption based on hard learning problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 595–618. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_35

    CrossRef  Google Scholar 

  3. Albrecht, M.R., Deo, A.: Large modulus ring-LWE \(\ge \) module-LWE. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8

    CrossRef  Google Scholar 

  4. Aggarwal, D., Joux, A., Prakash, A., Santha, M.: A new public-key cryptosystem via Mersenne numbers. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10993, pp. 459–482. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_16

    CrossRef  Google Scholar 

  5. Budroni, A., Chetioui, B., Franch, E.: Attacks on integer-RLWE. IACR Cryptol. ePrint Arch. 2020, 1007 (2020)

    Google Scholar 

  6. Bootland, C., Castryck, W., Szepieniec, A., Vercauteren, F.: A framework for cryptographic problems from linear algebra. J. Math. Cryptol. 14(1), 202–217 (2020)

    CrossRef  MathSciNet  Google Scholar 

  7. Boneh, D., Freeman, D.M.: Linearly homomorphic signatures over binary fields and new tools for lattice-based signatures. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 1–16. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19379-8_1

    CrossRef  Google Scholar 

  8. Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: ITCS 2012 (2012)

    Google Scholar 

  9. Bindel, N., Hamburg, M., Hövelmanns, K., Hülsing, A., Persichetti, E.: Tighter proofs of CCA security in the quantum random oracle model. In: Hofheinz, D., Rosen, A. (eds.) TCC 2019. LNCS, vol. 11892, pp. 61–90. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-36033-7_3

    CrossRef  MATH  Google Scholar 

  10. Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: STOC 2013 (2013)

    Google Scholar 

  11. Bai, S., Lepoint, T., Roux-Langlois, A., Sakzad, A., Stehlé, D., Steinfeld, R.: Improved security proofs in lattice-based cryptography: using the Rényi divergence rather than the statistical distance. J. Cryptol. 31(2), 610–640 (2018)

    CrossRef  Google Scholar 

  12. Bernstein, D.J., Persichetti, E.: Towards KEM unification. IACR Cryptol. ePrint Arch. 2018, 526 (2018)

    Google Scholar 

  13. Banerjee, A., Peikert, C., Rosen, A.: Pseudorandom functions and lattices. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 719–737. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_42

    CrossRef  Google Scholar 

  14. Gentry, C.: Key recovery and message attacks on NTRU-composite. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 182–194. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44987-6_12

    CrossRef  Google Scholar 

  15. Gu, C.: Integer version of ring-LWE and its applications. IACR Cryptol. ePrint Arch. 2017, 641 (2017)

    Google Scholar 

  16. Gu, C.: Integer version of ring-LWE and its applications. In: Meng, W., Furnell, S. (eds.) SocialSec 2019. CCIS, vol. 1095, pp. 110–122. Springer, Singapore (2019). https://doi.org/10.1007/978-981-15-0758-8_9

    CrossRef  Google Scholar 

  17. Hamburg, M.: Three bears: Candidate to NIS (2017)

    Google Scholar 

  18. Hofheinz, D., Hövelmanns, K., Kiltz, E.: A modular analysis of the Fujisaki-Okamoto transformation. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 341–371. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_12

    CrossRef  MATH  Google Scholar 

  19. Kuchta, V., Sakzad, A., Stehlé, D., Steinfeld, R., Sun, S.-F.: Measure-rewind-measure: tighter quantum random oracle model proofs for one-way to hiding and CCA security. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12107, pp. 703–728. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_24

    CrossRef  Google Scholar 

  20. Lyubashevsky, V., Micciancio, D.: Generalized compact knapsacks are collision resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 144–155. Springer, Heidelberg (2006). https://doi.org/10.1007/11787006_13

    CrossRef  Google Scholar 

  21. Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_1

    CrossRef  Google Scholar 

  22. Langlois, A., Stehlé, D.: Worst-case to average-case reductions for module lattices. Des. Codes Crypt. 75(3), 565–599 (2014). https://doi.org/10.1007/s10623-014-9938-4

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Langlois, A., Stehlé, D., Steinfeld, R.: GGHLite: more efficient multilinear maps from ideal lattices. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 239–256. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_14

    CrossRef  Google Scholar 

  24. NIST: Post-Quantum Cryptography Project. https://csrc.nist.gov/Projects/Post-Quantum-Cryptography/

  25. Peikert, C., Regev, O., Stephens-Davidowitz, N.: Pseudorandomness of ring-LWE for any ring and modulus. In: STOC 2017 (2017)

    Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

  27. Rosca, M., Stehlé, D., Wallet, A.: On the ring-LWE and polynomial-LWE problems. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 146–173. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_6

    CrossRef  Google Scholar 

  28. Stehlé, D., Steinfeld, R., Tanaka, K., Xagawa, K.: Efficient public key encryption based on ideal lattices. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 617–635. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10366-7_36

    CrossRef  Google Scholar 

  29. Saito, T., Xagawa, K., Yamakawa, T.: Tightly-secure key-encapsulation mechanism in the quantum random oracle model. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10822, pp. 520–551. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78372-7_17

    CrossRef  MATH  Google Scholar 

  30. Szepieniec, A.: Ramstake: Candidate to NIS (2017)

    Google Scholar 

Download references

Acknowledgments

This work was supported in part by European Union Horizon 2020 Research and Innovation Program Grant 780701, Australian Research Council Discovery Project Grant DP180102199, and by BPI-France in the context of the national project RISQ (P141580).

Author information

Authors and Affiliations

  1. ENS de Lyon, Laboratoire LIP (University of Lyon, CNRS, ENSL, INRIA, UCBL), Lyon, France

    Julien Devevey & Damien Stehlé

  2. Faculty of Information Technology, Monash University, Clayton, Australia

    Amin Sakzad & Ron Steinfeld

  3. Institut Universitaire de France, Paris, France

    Damien Stehlé

Authors
  1. Julien Devevey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amin Sakzad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Damien Stehlé
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ron Steinfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julien Devevey .

Editor information

Editors and Affiliations

  1. Texas A&M University, College Station, TX, USA

    Prof. Juan A. Garay

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 International Association for Cryptologic Research

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Devevey, J., Sakzad, A., Stehlé, D., Steinfeld, R. (2021). On the Integer Polynomial Learning with Errors Problem. In: Garay, J.A. (eds) Public-Key Cryptography – PKC 2021. PKC 2021. Lecture Notes in Computer Science(), vol 12710. Springer, Cham. https://doi.org/10.1007/978-3-030-75245-3_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-75245-3_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-75244-6

  • Online ISBN: 978-3-030-75245-3

  • eBook Packages: Computer ScienceComputer Science (R0)