Skip to main content
SpringerLink
Log in

Non-interactive Zero-Knowledge Functional Proofs

  • Conference paper
  • First Online:
Advances in Cryptology – ASIACRYPT 2023 (ASIACRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14442))

  • 428 Accesses

Abstract

In this paper, we consider to generalize NIZK by empowering a prover to share a witness in a fine-grained manner with verifiers. Roughly, the prover is able to authorize a verifier to obtain extra information of witness, i.e., besides verifying the truth of the statement, the verifier can additionally obtain certain function of the witness from the accepting proof using a secret functional key provided by the prover.

To fulfill these requirements, we introduce a new primitive called non-interactive zero-knowledge functional proofs (fNIZKs), and formalize its security notions. We provide a generic construction of fNIZK for any \({\textsf{NP}}\) relation \(\mathcal {R}\), which enables the prover to share any function of the witness with a verifier. For a widely-used relation about set membership proof (implying range proof), we construct a concrete and efficient fNIZK, through new building blocks (set membership encryption and dual inner-product encryption), which might be of independent interest.

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 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
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

Notes

  1. 1.

    A commitment scheme supports Sigma protocols, if there exists a Sigma protocol to prove the well-formedness of a commitment.

  2. 2.

    In this scenario, the authority generates a public key and a secret key, and all users utilize the public key to generate or verify proofs. The secret functional keys are generated by the authority, using the secret key.

  3. 3.

    For each \(x\in \mathcal {L}_{\mathcal {R}_{sm }}\), its witness is in the form of \((w,r_{\text {com} })\). In fSMP, we are only interested in functions of w (rather than \(r_{\text {com} }\)). So we define \(\mathbb {F}\) as family of functions whose domain is \(\mathcal {W}\) (rather than \(\mathcal {W}\times \mathcal{R}\mathcal{S}_{\textsf {Commit} .\textsf {Com} }\)).

  4. 4.

    Note that the Sigma protocol \(\mathrm{\Sigma }^{\mathcal {R}''}_{clause }\) supports that the verifier can recover the commitment via the responses and the challenge, so we can save the proof size by only sending the responds to the verifier as the proof.

References

  1. Blum, M., Feldman, P., Micali, S.: Non-interactive zero-knowledge and its applications (extended abstract). In: STOC 1988, pp. 103–112. ACM (1988)

    Google Scholar 

  2. Boneh, D., Sahai, A., Waters, B.: Functional encryption: definitions and challenges. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 253–273. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19571-6_16

    Chapter  Google Scholar 

  3. Boneh, D., Shoup, V.: A graduate course in applied cryptography. Draft 0.6 (2023)

    Google Scholar 

  4. Bunz, B., Bootle, J., Boneh, D., Poelstra, A., Wuille, P., Maxwell, G.: Bulletproofs: short proofs for confidential transactions and more. In: S &P 2018, pp. 315–334. IEEE (2018)

    Google Scholar 

  5. Camenisch, J., Chaabouni, R., Shelat, A.: Efficient protocols for set membership and range proofs. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 234–252. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-89255-7_15

    Chapter  Google Scholar 

  6. Camenisch, J., Hohenberger, S., Lysyanskaya, A.: Compact E-Cash. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 302–321. Springer, Heidelberg (2005). https://doi.org/10.1007/11426639_18

    Chapter  Google Scholar 

  7. Camenisch, J., Lysyanskaya, A.: An efficient system for non-transferable anonymous credentials with optional anonymity revocation. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 93–118. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44987-6_7

    Chapter  Google Scholar 

  8. Chaum, D., Pedersen, T.P.: Wallet databases with observers. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 89–105. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-48071-4_7

    Chapter  Google Scholar 

  9. Chaum, D., van Heyst, E.: Group signatures. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 257–265. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-46416-6_22

    Chapter  Google Scholar 

  10. Chen, J., Gay, R., Wee, H.: Improved dual system ABE in prime-order groups via predicate encodings. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 595–624. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_20

    Chapter  Google Scholar 

  11. Chen, J., Gong, J., Kowalczyk, L., Wee, H.: Unbounded ABE via bilinear entropy expansion, revisited. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 503–534. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_19

    Chapter  Google Scholar 

  12. Chen, J., Gong, J., Wee, H.: Improved inner-product encryption with adaptive security and full attribute-hiding. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018. LNCS, vol. 11273, pp. 673–702. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03329-3_23

    Chapter  Google Scholar 

  13. Couteau, G., Klooß, M., Lin, H., Reichle, M.: Efficient range proofs with transparent setup from bounded integer commitments. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12698, pp. 247–277. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77883-5_9

    Chapter  Google Scholar 

  14. Escala, A., Herold, G., Kiltz, E., Ràfols, C., Villar, J.: An algebraic framework for diffie-hellman assumptions. J. Cryptology 30, 242–288 (2017)

    Article  MathSciNet  Google Scholar 

  15. Feige, U., Lapidot, D., Shamir, A.: Multiple noninteractive zero knowledge proofs under general assumptions. SIAM J. Comput. 29(1), 1–28 (1999)

    Article  MathSciNet  Google Scholar 

  16. Fiat, A., Shamir, A.: How To prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_12

    Chapter  Google Scholar 

  17. Goel, A., Green, M., Hall-Andersen, M., Kaptchuk, G.: Stacking sigmas: a framework to Compose \(\Sigma \)-protocols for disjunctions. In: EUROCRYPT 2022, pp. 458–487. Springer (2022). https://doi.org/10.1007/978-3-031-07085-3_16

  18. Goldwasser, S.: How to play any mental game, or a completeness theorem for protocols with an honest majority. In: STOC 1987, pp. 218–229 (1987)

    Google Scholar 

  19. Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989)

    Article  MathSciNet  Google Scholar 

  20. Lewko, A., Okamoto, T., Sahai, A., Takashima, K., Waters, B.: Fully secure functional encryption: attribute-based encryption and (hierarchical) inner product encryption. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 62–91. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_4

    Chapter  Google Scholar 

  21. McCorry, P., Shahandashti, S.F., Hao, F.: A smart contract for boardroom voting with maximum voter privacy. In: Kiayias, A. (ed.) FC 2017. LNCS, vol. 10322, pp. 357–375. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70972-7_20

    Chapter  Google Scholar 

  22. Okamoto, T.: An efficient divisible electronic cash scheme. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 438–451. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-44750-4_35

    Chapter  Google Scholar 

  23. Okamoto, T., Takashima, K.: Hierarchical predicate encryption for inner-products. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 214–231. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10366-7_13

    Chapter  Google Scholar 

  24. Okamoto, T., Takashima, K.: Adaptively attribute-hiding (hierarchical) inner product encryption. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 591–608. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_35

    Chapter  Google Scholar 

  25. O’Neill, A.: Definitional issues in functional encryption. Cryptology ePrint Archive (2010)

    Google Scholar 

  26. 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). https://doi.org/10.1007/3-540-46766-1_9

    Chapter  Google Scholar 

  27. Sahai, A.: Non-malleable non-interactive zero knowledge and adaptive chosen-ciphertext security. In: FOCS 1999, pp. 543–553. IEEE Computer Society (1999)

    Google Scholar 

  28. Sasson, E.B., et al.: Zerocash: decentralized anonymous payments from bitcoin. In: S &P 2014, pp. 459–474. IEEE (2014)

    Google Scholar 

  29. Schnorr, C.P.: Efficient identification and signatures for smart cards. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 239–252. Springer, New York (1990). https://doi.org/10.1007/0-387-34805-0_22

    Chapter  Google Scholar 

  30. Tan, S.-Y., Groß, T.: MoniPoly—an expressive q-SDH-based anonymous attribute-based credential system. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12493, pp. 498–526. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64840-4_17

    Chapter  Google Scholar 

  31. Wee, H.: Attribute-hiding predicate encryption in bilinear groups, revisited. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 206–233. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_8

    Chapter  Google Scholar 

  32. Zhang, Y.: Introducing zkEVM (2022). https://scroll.io/blog/zkEVM

Download references

Acknowledgements

We would like to express our sincere appreciation to Junqing Gong for his valuable suggestions on the inner-product encryption! We also want to express our sincere appreciation to the anonymous reviewers for their valuable comments and suggestions! Gongxian Zeng and Zhengan Huang was supported by The Major Key Project of PCL (PCL2023A09). Junzuo Lai was supported by National Natural Science Foundation of China under Grant No. U2001205, Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023B1515040020), Industrial project No. TC20200930001. Jian Weng was supported by National Natural Science Foundation of China under Grant Nos. 61825203, 62332007 and U22B2028, Major Program of Guangdong Basic and Applied Research Project under Grant No. 2019B030302008, Guangdong Provincial Science and Technology Project under Grant No. 2021A0505030033, Science and Technology Major Project of Tibetan Autonomous Region of China under Grant No. XZ202201ZD0006G, National Joint Engineering Research Center of Network Security Detection and Protection Technology, Guangdong Key Laboratory of Data Security and Privacy Preserving, Guangdong Hong Kong Joint Laboratory for Data Security and Privacy Protection, and Engineering Research Center of Trustworthy AI, Ministry of Education. This research is supported by the National Research Foundation, Singapore under its Strategic Capability Research Centres Funding Initiative. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore.

Author information

Authors and Affiliations

  1. Peng Cheng Laboratory, Shenzhen, China

    Gongxian Zeng & Zhengan Huang

  2. College of Information Science and Technology, Jinan University, Guangzhou, China

    Junzuo Lai & Jian Weng

  3. Nanyang Technological University, Singapore, Singapore

    Linru Zhang, Xiangning Wang, Kwok-Yan Lam & Huaxiong Wang

Authors
  1. Gongxian Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junzuo Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhengan Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Linru Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiangning Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Kwok-Yan Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Huaxiong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Jian Weng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Junzuo Lai , Zhengan Huang , Linru Zhang or Xiangning Wang .

Editor information

Editors and Affiliations

  1. Nanyang Technological University, Singapore, Singapore

    Jian Guo

  2. Monash University, Melbourne, VIC, Australia

    Ron Steinfeld

Rights and permissions

Reprints and permissions

Copyright information

© 2023 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Zeng, G. et al. (2023). Non-interactive Zero-Knowledge Functional Proofs. In: Guo, J., Steinfeld, R. (eds) Advances in Cryptology – ASIACRYPT 2023. ASIACRYPT 2023. Lecture Notes in Computer Science, vol 14442. Springer, Singapore. https://doi.org/10.1007/978-981-99-8733-7_8

Download citation

  • DOI: https://doi.org/10.1007/978-981-99-8733-7_8

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-99-8732-0

  • Online ISBN: 978-981-99-8733-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation