Skip to main content
SpringerLink
Log in

Lattice-Based Proof of a Shuffle

  • Conference paper
  • First Online:
Financial Cryptography and Data Security (FC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11599))

Included in the following conference series:

Abstract

In this paper we present the first fully post-quantum proof of a shuffle for RLWE encryption schemes. Shuffles are commonly used to construct mixing networks (mix-nets), a key element to ensure anonymity in many applications such as electronic voting systems. They should preserve anonymity even against an attack using quantum computers in order to guarantee long-term privacy. The proof presented in this paper is built over RLWE commitments which are perfectly binding and computationally hiding under the RLWE assumption, thus achieving security in a post-quantum scenario. Furthermore we provide a new definition for a secure mixing node (mix-node) and prove that our construction satisfies this definition.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. Abe, M.: Universally verifiable mix-net with verification work independent of the number of mix-servers. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 437–447. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054144

    Chapter  Google Scholar 

  2. Abe, M.: Mix-networks on permutation networks. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 258–273. Springer, Heidelberg (1999). https://doi.org/10.1007/978-3-540-48000-6_21

    Chapter  Google Scholar 

  3. Abe, M., Hoshino, F.: Remarks on mix-network based on permutation networks. In: Kim, K. (ed.) PKC 2001. LNCS, vol. 1992, pp. 317–324. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44586-2_23

    Chapter  Google Scholar 

  4. Baum, C., Damgård, I., Lyubashevsky, V., Oechsner, S., Peikert, C.: More efficient commitments from structured lattice assumptions. In: Catalano, D., De Prisco, R. (eds.) SCN 2018. LNCS, vol. 11035, pp. 368–385. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98113-0_20

    Chapter  Google Scholar 

  5. Baum, C., Lyubashevsky, V.: Simple amortized proofs of shortness for linear relations over polynomial rings. Cryptology ePrint Archive, Report 2017/759 (2017). http://eprint.iacr.org/2017/759

  6. Bayer, S., Groth, J.: Zero-knowledge argument for polynomial evaluation with application to blacklists. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 646–663. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_38

    Chapter  Google Scholar 

  7. Benhamouda, F., Krenn, S., Lyubashevsky, V., Pietrzak, K.: Efficient zero-knowledge proofs for commitments from learning with errors over rings. In: Pernul, G., Ryan, P.Y.A., Weippl, E. (eds.) ESORICS 2015, Part I. LNCS, vol. 9326, pp. 305–325. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24174-6_16

    Chapter  Google Scholar 

  8. Bünz, B., Bootle, J., Boneh, D., Poelstra, A., Wuille, P., Maxwell, G.: Bulletproofs: short proofs for confidential transactions and more. In: 2018 IEEE Symposium on Security and Privacy, pp. 315–334. IEEE Computer Society Press, San Francisco (2018). https://doi.org/10.1109/SP.2018.00020

  9. Chaum, D.L.: Untraceable electronic mail, return addresses, and digital pseudonyms. Commun. ACM 24(2), 84–90 (1981). https://doi.org/10.1145/358549.358563

    Article  Google Scholar 

  10. Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: A homomorphic LWE based E-voting scheme. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 245–265. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29360-8_16

    Chapter  Google Scholar 

  11. Costa, N., Martínez, R., Morillo, P.: Proof of a shuffle for lattice-based cryptography. In: Lipmaa, H., Mitrokotsa, A., Matulevičius, R. (eds.) NordSec 2017. LNCS, vol. 10674, pp. 280–296. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70290-2_17

    Chapter  Google Scholar 

  12. Cramer, R., Damgård, I., Xing, C., Yuan, C.: Amortized complexity of zero-knowledge proofs revisited: achieving linear soundness slack. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part I. LNCS, vol. 10210, pp. 479–500. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_17

    Chapter  Google Scholar 

  13. Damgard, I.: On \(\sigma \)-protocols. Lecture on Cryptologic Protocol Theory. Faculty of Science, University of Aarhus (2010)

    Google Scholar 

  14. del Pino, R., Lyubashevsky, V.: Amortization with fewer equations for proving knowledge of small secrets. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part III. LNCS, vol. 10403, pp. 365–394. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63697-9_13

    Chapter  Google Scholar 

  15. del Pino, R., Lyubashevsky, V., Neven, G., Seiler, G.: Practical quantum-safe voting from lattices. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D. (eds.) ACM CCS 2017, pp. 1565–1581. ACM Press, Dallas (2017). https://doi.org/10.1145/3133956.3134101

  16. Furukawa, J.: Efficient and verifiable shuffling and shuffle-decryption. IEICE Trans. 88–A, 172–188 (2005)

    Article  Google Scholar 

  17. Furukawa, J., Sako, K.: An efficient scheme for proving a shuffle. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 368–387. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44647-8_22

    Chapter  Google Scholar 

  18. Groth, J.: A verifiable secret shuffe of homomorphic encryptions. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 145–160. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36288-6_11

    Chapter  Google Scholar 

  19. Groth, J., Ishai, Y.: Sub-linear zero-knowledge argument for correctness of a shuffle. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 379–396. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78967-3_22

    Chapter  Google Scholar 

  20. Groth, J., Lu, S.: Verifiable shuffle of large size ciphertexts. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 377–392. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71677-8_25

    Chapter  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

    Chapter  Google Scholar 

  22. Markus, J., Ari, J.: Millimix: mixing in small batches. Technical report, Center for Discrete Mathematics, Theoretical Computer Science (1999)

    Google Scholar 

  23. Micciancio, D., Regev, O.: Lattice-based cryptography. In: Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.) Post-Quantum Cryptography, pp. 147–191. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-540-88702-7

    Chapter  MATH  Google Scholar 

  24. Neff, C.A.: A verifiable secret shuffle and its application to e-voting. In: Reiter, M.K., Samarati, P. (eds.) ACM CCS 2001, pp. 116–125. ACM Press, Philadelphia (2001). https://doi.org/10.1145/501983.502000

  25. Neff, C.A.: Verifiable mixing (shuffling) of ElGamal pairs. VoteHere, Inc. (2003)

    Google Scholar 

  26. Peikert, C.: A decade of lattice cryptography. Cryptology ePrint Archive, Report 2015/939 (2015). http://eprint.iacr.org/2015/939

  27. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) 37th ACM STOC, pp. 84–93. ACM Press, Baltimore (2005). https://doi.org/10.1145/1060590.1060603

  28. Sako, K., Kilian, J.: Receipt-free mix-type voting scheme - a practical solution to the implementation of a voting booth. In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 393–403. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-49264-X_32

    Chapter  Google Scholar 

  29. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997). https://doi.org/10.1137/S0097539795293172

    Article  MathSciNet  MATH  Google Scholar 

  30. Singh, K., Pandu Rangan, C., Banerjee, A.K.: Lattice based mix network for location privacy in mobile system. Mob. Inf. Syst. 2015, 1–9 (2015). https://doi.org/10.1155/2015/963628

    Article  Google Scholar 

  31. Strand, M.: A verifiable shuffle for the GSW cryptosystem. In: Zohar, A., et al. (eds.) FC 2018. LNCS, vol. 10958, pp. 165–180. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-662-58820-8_12

    Chapter  Google Scholar 

  32. Terelius, B., Wikström, D.: Proofs of restricted shuffles. In: Bernstein, D.J., Lange, T. (eds.) AFRICACRYPT 2010. LNCS, vol. 6055, pp. 100–113. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12678-9_7

    Chapter  Google Scholar 

  33. Unruh, D.: Non-interactive zero-knowledge proofs in the quantum random oracle model. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part II. LNCS, vol. 9057, pp. 755–784. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_25

    Chapter  MATH  Google Scholar 

  34. Unruh, D.: Post-quantum security of Fiat-Shamir. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017, Part I. LNCS, vol. 10624, pp. 65–95. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_3

    Chapter  Google Scholar 

  35. Wikström, D.: The security of a mix-center based on a semantically secure cryptosystem. In: Menezes, A., Sarkar, P. (eds.) INDOCRYPT 2002. LNCS, vol. 2551, pp. 368–381. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36231-2_29

    Chapter  Google Scholar 

  36. Wikström, D.: A commitment-consistent proof of a shuffle. In: Boyd, C., González Nieto, J. (eds.) ACISP 2009. LNCS, vol. 5594, pp. 407–421. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02620-1_28

    Chapter  Google Scholar 

Download references

Acknowledgements

We would like to thank Kristian Gjøsteen for his helpful comments that greatly improved the proposal.

This work is partially supported by the European Union PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701) and the Spanish Ministry of Economy and Competitiveness, under Project MTM2016-77213-R.

Author information

Authors and Affiliations

  1. Scytl Secure Electronic Voting, Barcelona, Spain

    Nuria Costa

  2. Universitat Politècnica de Catalunya, Barcelona, Spain

    Ramiro Martínez & Paz Morillo

Authors
  1. Nuria Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ramiro Martínez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paz Morillo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ramiro Martínez .

Editor information

Editors and Affiliations

  1. Stirling University, Stirling, UK

    Andrea Bracciali

  2. Concordia University, Montréal, QC, Canada

    Jeremy Clark

  3. University of Oxford, Oxford, UK

    Federico Pintore

  4. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Peter B. Rønne

  5. University of Trento, Trento, Italy

    Massimiliano Sala

Rights and permissions

Reprints and permissions

Copyright information

© 2020 International Financial Cryptography Association

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Costa, N., Martínez, R., Morillo, P. (2020). Lattice-Based Proof of a Shuffle. In: Bracciali, A., Clark, J., Pintore, F., Rønne, P., Sala, M. (eds) Financial Cryptography and Data Security. FC 2019. Lecture Notes in Computer Science(), vol 11599. Springer, Cham. https://doi.org/10.1007/978-3-030-43725-1_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-43725-1_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43724-4

  • Online ISBN: 978-3-030-43725-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation