Skip to main content
Book cover

Theory of Cryptography Conference

TCC 2020: Theory of Cryptography pp 657–683Cite as

FHE-Based Bootstrapping of Designated-Prover NIZK

Part of the Lecture Notes in Computer Science book series (LNSC,volume 12550)

Abstract

We present a novel tree-based technique that can convert any designated-prover NIZK proof system (DP-NIZK) which maintains zero-knowledge only for single statement, into one that allows to prove an unlimited number of statements in ZK, while maintaining all parameters succinct. Our transformation requires leveled fully-homomorphic encryption. We note that single-statement DP-NIZK can be constructed from any one-way function. We also observe a two-way derivation between DP-NIZK and attribute-based signatures (ABS), and as a result derive now constructions of ABS and homomorphic signatures (HS).

Our construction improves upon the prior construction of lattice-based DP-NIZK by Kim and Wu (Crypto 2018) since we only require leveled FHE as opposed to HS (which also translates to improved LWE parameters when instantiated). Alternatively, the recent construction of NIZK without preprocessing from either circular-secure FHE (Canetti et al. STOC 2019) or polynomial Learning with Errors (Peikert and Shiehian, Crypto 2019) could be used to obtain a similar final statement. Nevertheless, we note that our statement is formally incomparable to these works (since leveled FHE is not known to imply circular secure FHE or the hardness of LWE). We view this as evidence for the potential in our technique, which we hope can find additional applications in future works.

Z. Brakerski and R. Tsabary—Supported by the Binational Science Foundation (Grant No. 2016726), and by the European Union Horizon 2020 Research and Innovation Program via ERC Project REACT (Grant 756482) and via Project PROMETHEUS (Grant 780701).

S. Garg—Supported in part from AFOSR Award FA9550-19-1-0200, NSF CNS Award 1936826, DARPA SIEVE Award, and research grants by the Sloan Foundation, Visa Inc., and Center for Long-Term Cybersecurity (CLTC, UC Berkeley). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agencies.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-64375-1_23
  • Chapter length: 27 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   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-64375-1
  • 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   109.99
Price excludes VAT (USA)
Learn about institutional subscriptions

References

  1. Boneh, D., Freeman, D.M.: Homomorphic signatures for polynomial functions. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 149–168. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_10

    CrossRef  Google Scholar 

  2. 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 

  3. Bellare, M., Fuchsbauer, G.: Policy-based signatures. In: Krawczyk [Kra14], pp. 520–537

    Google Scholar 

  4. Boneh, D., Freeman, D., Katz, J., Waters, B.: Signing a linear subspace: signature schemes for network coding. In: Jarecki, S., Tsudik, G. (eds.) PKC 2009. LNCS, vol. 5443, pp. 68–87. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00468-1_5

    CrossRef  Google Scholar 

  5. Blum, M., Feldman, P., Micali, S.: Non-interactive zero-knowledge and its applications (extended abstract). In: Simon, J. (ed.) Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 2–4 May 1988, Chicago, Illinois, USA, pp. 103–112. ACM (1988)

    Google Scholar 

  6. Boyle, E., Goldwasser, S., Ivan, I.: Functional signatures and pseudorandom functions. In: Krawczyk [Kra14], pp. 501–519. IACR ePrint (2013). http://eprint.iacr.org/2013/401

  7. Brakerski, Z., Vaikuntanathan, V.: Lattice-based FHE as secure as PKE. In: Naor, M. (ed.) Innovations in Theoretical Computer Science, ITCS 2014, Princeton, NJ, USA, 12–14 January 2014, pp. 1–12. ACM (2014)

    Google Scholar 

  8. Boneh, D., Zhandry, M.: Multiparty key exchange, efficient traitor tracing, and more from indistinguishability obfuscation. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 480–499. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_27

    CrossRef  Google Scholar 

  9. Canetti, R., et al.: from practice to theory. In: Charikar, M., Cohen, E. (eds.) Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, 23–26 June 2019, pp. 1082–1090. ACM (2019)

    Google Scholar 

  10. Charles, D.X., Jain, K., Lauter, K.E.: Signatures for network coding. IJICoT 1(1), 3–14 (2009)

    Google Scholar 

  11. Damgård, I.: Non-interactive circuit based proofs and non-interactive perfect zero-knowledge with preprocessing. In: Rueppel, R.A. (ed.) EUROCRYPT 1992. LNCS, vol. 658, pp. 341–355. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-47555-9_28

    CrossRef  Google Scholar 

  12. Feige, U., Lapidot, D., Shamir, A.: Multiple non-interactive zero knowledge proofs based on a single random string (extended abstract). In: 31st Annual Symposium on Foundations of Computer Science, St. Louis, Missouri, USA, 22–24 October 1990, vol. I, pp. 308–317. IEEE Computer Society (1990)

    Google Scholar 

  13. Gentry, C.: A fully homomorphic encryption scheme. PhD thesis, Stanford University (2009). crypto.stanford.edu/craig

  14. Gentry, C., Groth, J., Ishai, Y., Peikert, C., Sahai, A., Smith, A.D.: Using fully homomorphic hybrid encryption to minimize non-interactive zero-knowledge proofs. J. Cryptol. 28(4), 820–843 (2015)

    CrossRef  Google Scholar 

  15. Gennaro, R., Katz, J., Krawczyk, H., Rabin, T.: Secure network coding over the integers. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 142–160. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_9

    CrossRef  Google Scholar 

  16. Gorbunov, S., Vaikuntanathan, V., Wichs, D.: Leveled fully homomorphic signatures from standard lattices. In: Servedio, R.A., Rubinfeld, R. (eds.) Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, 14–17 June 2015, pp. 469–477. ACM (2015)

    Google Scholar 

  17. Katsumata, S., Nishimaki, R., Yamada, S., Yamakawa, T.: Designated verifier/prover and preprocessing NIZKs from Diffie-Hellman assumptions. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11477, pp. 622–651. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17656-3_22

    CrossRef  Google Scholar 

  18. Katsumata, S., Nishimaki, R., Yamada, S., Yamakawa, T.: Compact NIZKs from standard assumptions on bilinear maps. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12107, pp. 379–409. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_13

    CrossRef  Google Scholar 

  19. Kilian, J., Petrank, E.: An efficient non-interactive zero-knowledge proof system for NP with general assumptions. Electron. Colloq. Comput. Complex. 2(38) (1995)

    Google Scholar 

  20. Krawczyk, H. (ed.): PKC 2014. LNCS, vol. 8383. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54631-0

    CrossRef  MATH  Google Scholar 

  21. Kim, S., Wu, D.J.: Multi-theorem preprocessing NIZKs from lattices. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 733–765. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_25

    CrossRef  Google Scholar 

  22. Maji, H.K., Prabhakaran, M., Rosulek, M.: Attribute-based signatures. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 376–392. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19074-2_24

    CrossRef  Google Scholar 

  23. Peikert, C., Shiehian, S.: Noninteractive zero knowledge for NP from (plain) learning with errors. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11692, pp. 89–114. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_4

    CrossRef  Google Scholar 

  24. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) STOC, pp. 84–93. ACM (2005). Full version in [?]

    Google Scholar 

  25. Sakai, Y., Attrapadung, N., Hanaoka, G.: Attribute-based signatures for circuits from bilinear map. In: Cheng, C.-M., Chung, K.-M., Persiano, G., Yang, B.-Y. (eds.) PKC 2016. LNCS, vol. 9614, pp. 283–300. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49384-7_11

    CrossRef  Google Scholar 

  26. Sakai, Y., Katsumata, S., Attrapadung, N., Hanaoka, G.: Attribute-based signatures for unbounded languages from standard assumptions. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018. LNCS, vol. 11273, pp. 493–522. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03329-3_17

    CrossRef  Google Scholar 

  27. De Santis, A., Micali, S., Persiano, G.: Non-interactive zero-knowledge proof systems. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 52–72. Springer, Heidelberg (1988). https://doi.org/10.1007/3-540-48184-2_5

    CrossRef  Google Scholar 

  28. Tsabary, R.: An equivalence between attribute-based signatures and homomorphic signatures, and new constructions for both. IACR Cryptology ePrint Archive 2017:723 (2017)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Weizmann Institute of Science, Rehovot, Israel

    Zvika Brakerski & Rotem Tsabary

  2. University of California, Berkeley, Berkeley, USA

    Sanjam Garg

Authors
  1. Zvika Brakerski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sanjam Garg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rotem Tsabary
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rotem Tsabary .

Editor information

Editors and Affiliations

  1. Cornell Tech, New York, NY, USA

    Prof. Rafael Pass

  2. Institute of Science and Technology Austria, Klosterneuburg, Austria

    Krzysztof Pietrzak

Rights and permissions

Reprints and Permissions

Copyright information

© 2020 International Association for Cryptologic Research

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Brakerski, Z., Garg, S., Tsabary, R. (2020). FHE-Based Bootstrapping of Designated-Prover NIZK. In: Pass, R., Pietrzak, K. (eds) Theory of Cryptography. TCC 2020. Lecture Notes in Computer Science(), vol 12550. Springer, Cham. https://doi.org/10.1007/978-3-030-64375-1_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-64375-1_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-64374-4

  • Online ISBN: 978-3-030-64375-1

  • eBook Packages: Computer ScienceComputer Science (R0)