Skip to main content
Springer Nature Link
Log in

Succinct Verification of Compressed Sigma Protocols in the Updatable SRS Setting

  • Conference paper
  • First Online:
Public-Key Cryptography – PKC 2024 (PKC 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14602))

Included in the following conference series:

  • 546 Accesses

Abstract

We propose protocols in the Compressed Sigma Protocol framework that achieve a succinct verifier. Towards this, we construct a new inner product argument and cast it in the Compressed Sigma Protocol (CSP) framework as a protocol for opening a committed linear form, achieving logarithmic verification.

We then use our succinct-verifier CSP to construct a zero-knowledge argument for circuit satisfiability (under the discrete logarithm assumption in bilinear groups) in the updatable Structured Reference String (SRS) setting that achieves \(O(\log n)\) proof size and \(O(\log n)\) verification complexity. Our circuit zero-knowledge protocol has concretely better proof/prover/verifier complexity compared to the state-of-the-art protocol in the updatable setting under the same assumption. Our techniques of achieving verifier-succinctness in the compression framework is of independent interest.

We then show a commitment scheme for committing to group elements using a structured commitment key. We construct protocols to open a committed homomorphism on a committed vector with verifier succinctness in the designated verifier setting. This has applications in making the verifier in compressed sigma protocols for bilinear group arithmetic circuits, succinct.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    We note that other SNARKs in the universal, updatable setting that have better communication and/verifier complexity \((O_\lambda (1))\) rely on the Algebraic Group Model or Knowledge Type assumptions in addition to the Random Oracle Model. In this work, we are interested in constructions in the Random Oracle Model, and relying on standard assumptions.

  2. 2.

    We note that the distribution remains the same even when \({\overline{{\textbf {a}}}}g_1\) is sampled from \(\mathcal{M}\mathcal{L}_{n_0}(\mathbb {G}_1)\) and g is then set to \(g=e(g_1,H)\), making the final commitment key for \(\mathbb {Z}_q\)-vector to be \(\textbf{g}={\overline{{\textbf {a}}}}g\), as opposed to when \(\textbf{g}\) is directly sampled from \(\mathcal{M}\mathcal{L}_{n_0}(\mathbb {G}_T)\).

References

  1. https://www.zellic.io/blog/zk-friendly-hash-functions

  2. Abe, M., Fuchsbauer, G., Groth, J., Haralambiev, K., Ohkubo, M.: Structure-preserving signatures and commitments to group elements. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 209–236. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_12

    Chapter  Google Scholar 

  3. Albrecht, M., Grassi, L., Rechberger, C., Roy, A., Tiessen, T.: MiMC: efficient encryption and cryptographic hashing with minimal multiplicative complexity. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016, Part I. LNCS, vol. 10031, pp. 191–219. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_7

    Chapter  Google Scholar 

  4. Arun, A., Ganesh, C., Lokam, S., Mopuri, T., Sridhar, S.: Dew: a transparent constant-sized polynomial commitment scheme. In: Boldyreva, A., Kolesnikov, V. (eds.) PKC 2023, Part II. LNCS, vol. 13941, pp. 542–571. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-31371-4_19

    Chapter  Google Scholar 

  5. Attema, T., Cramer, R.: Compressed \( \Sigma \)-protocol theory and practical application to plug  & play secure algorithmics. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020, Part III. LNCS, vol. 12172, pp. 513–543. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56877-1_18

    Chapter  Google Scholar 

  6. Attema, T., Cramer, R., Kohl, L.: A Compressed \( \Sigma \)-protocol theory for lattices. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021, Part II. LNCS, vol. 12826, pp. 549–579. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84245-1_19

    Chapter  Google Scholar 

  7. Attema, T., Cramer, R., Rambaud, M.: Compressed \(\Sigma \)-protocols for bilinear group arithmetic circuits and application to logarithmic transparent threshold signatures. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021, Part IV. LNCS, vol. 13093, pp. 526–556. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92068-5_18

    Chapter  Google Scholar 

  8. Bayer, S., Groth, J.: Efficient zero-knowledge argument for correctness of a shuffle. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 263–280. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_17

    Chapter  Google Scholar 

  9. Ben-Sasson, E., Chiesa, A., Tromer, E., Virza, M.: Succinct non-interactive zero knowledge for a von neumann architecture. In: Fu, K., Jung, J. (eds.) USENIX Security 2014, pp. 781–796. USENIX Association, August 2014

    Google Scholar 

  10. Bitansky, N., Chiesa, A., Ishai, Y., Paneth, O., Ostrovsky, R.: Succinct non-interactive arguments via linear interactive proofs. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 315–333. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36594-2_18

    Chapter  Google Scholar 

  11. Bootle, J., Cerulli, A., Chaidos, P., Groth, J., Petit, C.: Efficient zero-knowledge arguments for arithmetic circuits in the discrete log setting. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part II. LNCS, vol. 9666, pp. 327–357. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_12

    Chapter  Google Scholar 

  12. 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 (SP), pp. 315–334. IEEE (2018)

    Google Scholar 

  13. Bünz, B., Fisch, B., Szepieniec, A.: Transparent SNARKs from DARK compilers. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020, Part I. LNCS, vol. 12105, pp. 677–706. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45721-1_24

    Chapter  Google Scholar 

  14. Chiesa, A., Hu, Y., Maller, M., Mishra, P., Vesely, N., Ward, N.: Marlin: preprocessing zkSNARKs with universal and updatable SRS. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020, Part I. LNCS, vol. 12105, pp. 738–768. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45721-1_26

    Chapter  Google Scholar 

  15. Daza, V., Ràfols, C., Zacharakis, A.: Updateable inner product argument with logarithmic verifier and applications. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020, Part I. LNCS, vol. 12110, pp. 527–557. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45374-9_18

    Chapter  Google Scholar 

  16. Dutta, M., Ganesh, C., Jawalkar, N.: Succinct verification of compressed sigma protocols in the updatable SRS setting. Cryptology ePrint Archive, Paper 2024/075 (2024). https://eprint.iacr.org/2024/075

  17. 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 

  18. Gabizon, A., Williamson, Z.J., Ciobotaru, O.: Plonk: permutations over lagrange-bases for oecumenical noninteractive arguments of knowledge. Cryptology ePrint Archive, Report 2019/953 (2019). https://ia.cr/2019/953

  19. Gennaro, R., Gentry, C., Parno, B., Raykova, M.: Quadratic span programs and succinct NIZKs without PCPs. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 626–645. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_37

    Chapter  Google Scholar 

  20. 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 

  21. Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: Juels, A., Wright, R.N., De Capitani di Vimercati, S. (eds.) ACM CCS 2006, pp. 89–98. ACM Press, October/November 2006. Available as Cryptology ePrint Archive Report 2006/309

    Google Scholar 

  22. Groth, J.: Short pairing-based non-interactive zero-knowledge arguments. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 321–340. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17373-8_19

    Chapter  Google Scholar 

  23. Groth, J.: On the size of pairing-based non-interactive arguments. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part II. LNCS, vol. 9666, pp. 305–326. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_11

    Chapter  Google Scholar 

  24. Groth, J., Kohlweiss, M., Maller, M., Meiklejohn, S., Miers, I.: Updatable and universal common reference strings with applications to zk-SNARKs. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part III. LNCS, vol. 10993, pp. 698–728. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_24

    Chapter  Google Scholar 

  25. Kilian, J.: A note on efficient zero-knowledge proofs and arguments. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pp. 723–732 (1992)

    Google Scholar 

  26. Lai, R.W.F., Malavolta, G., Ronge, V.: Succinct arguments for bilinear group arithmetic: practical structure-preserving cryptography. In: Cavallaro, L., Kinder, J., Wang, X.F., Katz, J. (eds.) ACM CCS 2019, pp. 2057–2074. ACM Press, November 2019

    Google Scholar 

  27. Lee, J.: Dory: efficient, transparent arguments for generalised inner products and polynomial commitments. In: Nissim, K., Waters, B. (eds.) TCC 2021. LNCS, vol. 13043, pp. 1–34. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90453-1_1

    Chapter  Google Scholar 

  28. Lipmaa, H.: Progression-free sets and sublinear pairing-based non-interactive zero-knowledge arguments. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 169–189. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28914-9_10

    Chapter  Google Scholar 

  29. Lipmaa, H.: Succinct non-interactive zero knowledge arguments from span programs and linear error-correcting codes. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part I. LNCS, vol. 8269, pp. 41–60. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42033-7_3

    Chapter  Google Scholar 

  30. Lipmaa, H., Siim, J., Zajac, M.: Counting vampires: from univariate sumcheck to updatable ZK-SNARK. In: Agrawal, S., Lin, D. (eds.) ASIACRYPT 2022, Part II. LNCS, vol. 13792, pp. 249–278. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-22966-4_9

    Chapter  Google Scholar 

  31. Maller, M., Bowe, S., Kohlweiss, M., Meiklejohn, S.: Sonic: zero-knowledge SNARKs from linear-size universal and updatable structured reference strings. In: Cavallaro, L., Kinder, J., Wang, X.F., Katz, J. (eds.) ACM CCS 2019, pp. 2111–2128. ACM Press, November 2019

    Google Scholar 

  32. Micali, S.: CS proofs. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 436–453. IEEE (1994)

    Google Scholar 

  33. Parno, B., Howell, J., Gentry, C., Raykova, M.: Pinocchio: nearly practical verifiable computation. In: 2013 IEEE Symposium on Security and Privacy, pp. 238–252. IEEE Computer Society Press, May 2013

    Google Scholar 

  34. Sahai, A., Waters, B.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005). https://doi.org/10.1007/11426639_27

    Chapter  Google Scholar 

Download references

Acknowledgement

The research of the second author was supported by Core Research Grant CRG/2020/004488, SERB, Department of Science and Technology, India.

Author information

Authors and Affiliations

  1. Indian Institute of Science, Bangalore, India

    Moumita Dutta, Chaya Ganesh & Neha Jawalkar

Authors
  1. Moumita Dutta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chaya Ganesh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Neha Jawalkar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moumita Dutta .

Editor information

Editors and Affiliations

  1. The University of Sydney, Sydney, NSW, Australia

    Qiang Tang

  2. The Australian National University, Acton, ACT, Australia

    Vanessa Teague

Rights and permissions

Reprints and permissions

Copyright information

© 2024 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dutta, M., Ganesh, C., Jawalkar, N. (2024). Succinct Verification of Compressed Sigma Protocols in the Updatable SRS Setting. In: Tang, Q., Teague, V. (eds) Public-Key Cryptography – PKC 2024. PKC 2024. Lecture Notes in Computer Science, vol 14602. Springer, Cham. https://doi.org/10.1007/978-3-031-57722-2_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-57722-2_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-57721-5

  • Online ISBN: 978-3-031-57722-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships