Skip to main content
SpringerLink
Log in

Proof of a Shuffle for Lattice-Based Cryptography

  • Conference paper
  • First Online:
Secure IT Systems (NordSec 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10674))

Included in the following conference series:

Abstract

In this paper we present the first proof of a shuffle for lattice-based cryptography which can be used to build a universally verifiable mix-net capable of mixing votes encrypted with a post-quantum algorithm, thus achieving long-term privacy. Universal verifiability is achieved by means of the publication of a non-interactive zero knowledge proof of a shuffle generated by each mix-node which can be verified by any observer. This published data guarantees long-term privacy since its security is based on perfectly hiding commitments and also on the hardness of solving the Ring Learning With Errors (RLWE) problem, that is widely believed to be quantum resistant.

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.: 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). doi:10.1007/978-3-540-48000-6_21

    Chapter  Google Scholar 

  2. Adida, B., Wikström, D.: How to shuffle in public. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 555–574. Springer, Heidelberg (2007). doi:10.1007/978-3-540-70936-7_30

    Chapter  Google Scholar 

  3. Adida, B., Wikström, D.: Offline/Online mixing. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 484–495. Springer, Heidelberg (2007). doi:10.1007/978-3-540-73420-8_43

    Chapter  Google Scholar 

  4. 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). doi:10.1007/978-3-642-03356-8_35

    Chapter  Google Scholar 

  5. 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. LNCS, vol. 9326, pp. 305–325. Springer, Cham (2015). doi:10.1007/978-3-319-24174-6_16

    Chapter  Google Scholar 

  6. Buchmann, J., Demirel, D., Graaf, J.: Towards a publicly-verifiable mix-net providing everlasting privacy. In: Sadeghi, A.-R. (ed.) FC 2013. LNCS, vol. 7859, pp. 197–204. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39884-1_16

    Chapter  Google Scholar 

  7. Canetti, R.: Universally composable security: a new paradigm for cryptographic protocols. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, FOCS 2001, Washington, USA, pp. 136–145. IEEE Computer Society (2001)

    Google Scholar 

  8. Chaum, D.L.: Untraceable electronic mail, return addresses, and digital pseudonyms. Commun. ACM 24(2), 84–90 (1981)

    Article  Google Scholar 

  9. 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). doi:10.1007/978-3-319-29360-8_16

    Chapter  Google Scholar 

  10. Costa, N., Martínez, R., Morillo, P.: Proof of a shuffle for lattice-based cryptography. IACR Cryptology ePrint Archive (2017)

    Google Scholar 

  11. Cramer, R., Gennaro, R., Schoenmakers, B.: A secure and optimally efficient multi-authority election scheme. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 103–118. Springer, Heidelberg (1997). doi:10.1007/3-540-69053-0_9

    Chapter  Google Scholar 

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

    Google Scholar 

  13. Demirel, D., Henning, M., van de Graaf, J., Ryan, P.Y.A., Buchmann, J.: Prêt à voter providing everlasting privacy. In: Heather, J., Schneider, S., Teague, V. (eds.) Vote-ID 2013. LNCS, vol. 7985, pp. 156–175. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39185-9_10

    Chapter  Google Scholar 

  14. Fauzi, P., Lipmaa, H.: Efficient culpably sound NIZK shuffle argument without random oracles. In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 200–216. Springer, Cham (2016). doi:10.1007/978-3-319-29485-8_12

    Chapter  Google Scholar 

  15. Fauzi, P., Lipmaa, H., Zając, M.: A shuffle argument secure in the generic model. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10032, pp. 841–872. Springer, Heidelberg (2016). doi:10.1007/978-3-662-53890-6_28

    Chapter  Google Scholar 

  16. 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). doi:10.1007/3-540-44647-8_22

    Chapter  Google Scholar 

  17. Golle, P., Jakobsson, M., Juels, A., Syverson, P.: Universal re-encryption for mixnets. In: Okamoto, T. (ed.) CT-RSA 2004. LNCS, vol. 2964, pp. 163–178. Springer, Heidelberg (2004). doi:10.1007/978-3-540-24660-2_14

    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). doi:10.1007/3-540-36288-6_11

    Chapter  Google Scholar 

  19. Groth, J., Lu, S.: A non-interactive shuffle with pairing based verifiability. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 51–67. Springer, Heidelberg (2007). doi:10.1007/978-3-540-76900-2_4

    Chapter  Google Scholar 

  20. 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). doi:10.1007/978-3-642-19074-2_21

    Chapter  Google Scholar 

  21. Ling, S., Nguyen, K., Stehlé, D., Wang, H.: Improved zero-knowledge proofs of knowledge for the ISIS problem, and applications. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 107–124. Springer, Heidelberg (2013). doi:10.1007/978-3-642-36362-7_8

    Chapter  Google Scholar 

  22. Lipmaa, H., Zhang, B.: A more efficient computationally sound non-interactive zero-knowledge shuffle argument. In: Visconti, I., Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 477–502. Springer, Heidelberg (2012). doi:10.1007/978-3-642-32928-9_27

    Chapter  Google Scholar 

  23. Locher, P., Haenni, R.: Verifiable internet elections with everlasting privacy and minimal trust. In: Haenni, R., Koenig, R.E., Wikström, D. (eds.) VOTELID 2015. LNCS, vol. 9269, pp. 74–91. Springer, Cham (2015). doi:10.1007/978-3-319-22270-7_5

    Chapter  Google Scholar 

  24. Locher, P., Haenni, R., Koenig, R.E.: Coercion-resistant internet voting with everlasting privacy. In: Clark, J., Meiklejohn, S., Ryan, P.Y.A., Wallach, D., Brenner, M., Rohloff, K. (eds.) FC 2016. LNCS, vol. 9604, pp. 161–175. Springer, Heidelberg (2016). doi:10.1007/978-3-662-53357-4_11

    Chapter  Google Scholar 

  25. Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. J. ACM 60(6), 43:1–43:35 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Markus, J., Ari, J.: Millimix: mixing in small batches. Technical report (1999)

    Google Scholar 

  27. 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). doi:10.1007/978-3-540-88702-7_5

    Chapter  Google Scholar 

  28. Moran, T., Naor, M.: Split-ballot voting: everlasting privacy with distributed trust. In: Proceedings of the 14th ACM Conference on Computer and Communications Security, CCS 2007, pp. 246–255. ACM (2007)

    Google Scholar 

  29. Andrew Neff, C.: A verifiable secret shuffle and its application to e-voting. In: Proceedings of the 8th ACM Conference on Computer and Communication Security, CCS 2001, pp. 116–125, NY, USA (2001)

    Google Scholar 

  30. Park, C., Itoh, K., Kurosawa, K.: Efficient anonymous channel and all/nothing election scheme. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 248–259. Springer, Heidelberg (1994). doi:10.1007/3-540-48285-7_21

    Chapter  Google Scholar 

  31. Pedersen, T.P.: Non-interactive and information-theoretic secure verifiable secret sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992). doi:10.1007/3-540-46766-1_9

    Google Scholar 

  32. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, STOC 2005, pp. 84–93, New York, NY, USA. ACM (2005)

    Google Scholar 

  33. Regev, O.: The learning with errors problem. In: IEEE 25th Annual Conference on Computational Complexity (CCC), pp. 191–204 (2010)

    Google Scholar 

  34. Rückert, M., Schneider, M.: Estimating the security of lattice-based cryptosystems. IACR Cryptology ePrint Archive, Report 2010/137 (2010). http://eprint.iacr.org/2010/137

  35. Sako, K., Kilian, J.: Receipt-free mix-type voting scheme. In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 393–403. Springer, Heidelberg (1995). doi:10.1007/3-540-49264-X_32

    Chapter  Google Scholar 

  36. Singh, K., Pandu Rangan, C., Banerjee, A.K.: Lattice based universal re-encryption for mixnet. J. Int. Serv. Inf. Secur. (JISIS) 4(1), 1–11 (2014)

    Google Scholar 

  37. Singh, K., Pandu Rangan, C., Banerjee, A.K.: Lattice based mix network for location privacy in mobile system. Mob. Inf. Syst. 1–9, 2015 (2015)

    Google Scholar 

  38. 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). doi:10.1007/978-3-642-12678-9_7

    Chapter  Google Scholar 

  39. 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). doi:10.1007/3-540-36231-2_29

    Chapter  Google Scholar 

  40. Wikström, D.: A universally composable mix-net. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 317–335. Springer, Heidelberg (2004). doi:10.1007/978-3-540-24638-1_18

    Chapter  Google Scholar 

  41. Wikström, D.: A sender verifiable mix-net and a new proof of a shuffle. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 273–292. Springer, Heidelberg (2005). doi:10.1007/11593447_15

    Chapter  Google Scholar 

  42. 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). doi:10.1007/978-3-642-02620-1_28

    Chapter  Google Scholar 

Download references

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 Nuria Costa .

Editor information

Editors and Affiliations

  1. University of Tartu, Tartu, Estonia

    Helger Lipmaa

  2. Chalmers University of Technology, Gothenburg, Sweden

    Aikaterini Mitrokotsa

  3. University of Tartu, Tartu, Estonia

    Raimundas Matulevičius

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Costa, N., Martínez, R., Morillo, P. (2017). Proof of a Shuffle for Lattice-Based Cryptography. In: Lipmaa, H., Mitrokotsa, A., Matulevičius, R. (eds) Secure IT Systems. NordSec 2017. Lecture Notes in Computer Science(), vol 10674. Springer, Cham. https://doi.org/10.1007/978-3-319-70290-2_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-70290-2_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-70289-6

  • Online ISBN: 978-3-319-70290-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation