Skip to main content
Springer Nature Link
Log in

Malleable Commitments from Group Actions and Zero-Knowledge Proofs for Circuits Based on Isogenies

  • Conference paper
  • First Online:
Progress in Cryptology – INDOCRYPT 2023 (INDOCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14459))

Included in the following conference series:

Abstract

Zero-knowledge proofs for NP statements are an essential tool for building various cryptographic primitives and have been extensively studied in recent years. In a seminal result from Goldreich, Micali and Wigderson [17], zero-knowledge proofs for NP statements can be built from any one-way function, but this construction leads very inefficient proofs. To yield practical constructions, one often uses the additional structure provided by homomorphic commitments.

In this paper, we introduce a relaxed notion of homomorphic commitments, called malleable commitments, which requires less structure to be instantiated. We provide a malleable commitment construction from the ElGamal-type isogeny-based group action from Eurocrypt’22 [5]. We show how malleable commitments with a group structure in the malleability can be used to build zero-knowledge proofs for NP statements, improving on the naive construction from one-way functions. We compare three different approaches, namely from arithmetic circuits, rank-1 constraint systems and branching programs.

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
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
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

Similar content being viewed by others

References

  1. Alamati, N., De Feo, L., Montgomery, H., Patranabis, S.: Cryptographic group actions and applications. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020, Part II. LNCS, vol. 12492, pp. 411–439. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_14

    Chapter  Google Scholar 

  2. Ames, S., Hazay, C., Ishai, Y., Venkitasubramaniam, M.: Ligero: lightweight sublinear arguments without a trusted setup. In: ACM CCS 2017, pp. 2087–2104 (2017)

    Google Scholar 

  3. Barrington, D.A.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC. In: Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, pp. 1–5 (1986)

    Google Scholar 

  4. Ben-Sasson, E., Chiesa, A., Riabzev, M., Spooner, N., Virza, M., Ward, N.P.: Aurora: transparent succinct arguments for R1CS. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019, Part I. LNCS, vol. 11476, pp. 103–128. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17653-2_4

    Chapter  Google Scholar 

  5. Beullens, W., Dobson, S., Katsumata, S., Lai, Y.-F., Pintore, F.: Group signatures and more from isogenies and lattices: generic, simple, and efficient. In: Dunkelman, O., Dziembowski, S. (eds.) EUROCRYPT 2022, Part II. LNCS, vol. 13276, pp. 95–126. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-07085-3_4

    Chapter  Google Scholar 

  6. Beullens, W., Katsumata, S., Pintore, F.: Calamari and Falafl: logarithmic (linkable) ring signatures from isogenies and lattices. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020, Part II. LNCS, vol. 12492, pp. 464–492. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_16

    Chapter  Google Scholar 

  7. Beullens, W., Kleinjung, T., Vercauteren, F.: CSI-FiSh: efficient isogeny based signatures through class group computations. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019, Part I. LNCS, vol. 11921, pp. 227–247. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34578-5_9

    Chapter  Google Scholar 

  8. Beullens, W., Seiler, G.: LABRADOR: compact proofs for R1CS from module-SIS. Cryptology ePrint Archive, Paper 2022/1341 (2023)

    Google Scholar 

  9. Boneh, D., Gentry, C., Gorbunov, S., Halevi, S., Nikolaenko, V., Segev, G., Vaikuntanathan, V., Vinayagamurthy, D.: Fully key-homomorphic encryption, arithmetic circuit ABE and compact garbled circuits. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 533–556. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_30

    Chapter  Google Scholar 

  10. 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 

  11. 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 (2018)

    Google Scholar 

  12. Cramer, R., Damgård, I., Schoenmakers, B.: Proofs of partial knowledge and simplified design of witness hiding protocols. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 174–187. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-48658-5_19

    Chapter  Google Scholar 

  13. De Feo, L., et al.: SCALLOP: scaling the CSI-FiSh. In: Boldyreva, A., Kolesnikov, V. (eds.) PKC 2023, Part I. LNCS, vol. 13940, pp. 345–375. Springer, Cham (2023)

    Chapter  Google Scholar 

  14. De Feo, L., Meyer, M.: Threshold schemes from isogeny assumptions. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020, Part II. LNCS, vol. 12111, pp. 187–212. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45388-6_7

    Chapter  Google Scholar 

  15. 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 

  16. Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_5

    Chapter  Google Scholar 

  17. Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in np have zero-knowledge proof systems. J. ACM 38, 691–729 (1991)

    Article  MathSciNet  Google Scholar 

  18. Gorbunov, S., Vaikuntanathan, V., Wichs, D.: Leveled fully homomorphic signatures from standard lattices. In: Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, New York, NY, USA, pp. 469–477 (2015)

    Google Scholar 

  19. Groth, J.: Homomorphic trapdoor commitments to group elements. Cryptology ePrint Archive, Paper 2009/007 (2009)

    Google Scholar 

  20. Groth, J.: Linear algebra with sub-linear zero-knowledge arguments. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 192–208. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_12

    Chapter  Google Scholar 

  21. Katsumata, S., Lai, Y., LeGrow, J.T., Qin, L.: CSI-otter: isogeny-based (partially) blind signatures from the class group action with a twist. In: Handschuh, H., Lysyanskaya, A. (eds.) CRYPTO 2023. LNCS, vol. 14083, pp. 729–761. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-38548-3_24

    Chapter  Google Scholar 

  22. Lai, Y.-F.: CAPYBARA and TSUBAKI: verifiable random functions from group actions and isogenies. Cryptology ePrint Archive, Report 2023/182 (2023)

    Google Scholar 

  23. 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 

  24. Peikert, Chris: He gives C-sieves on the CSIDH. In: Canteaut, Anne, Ishai, Yuval (eds.) EUROCRYPT 2020, Part II. LNCS, vol. 12106, pp. 463–492. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45724-2_16

    Chapter  Google Scholar 

  25. Wahby, R.S., Tzialla, I., shelat, A., Thaler, J., Walfish, M.: Doubly-efficient zkSNARKs without trusted setup. In: 2018 IEEE Symposium on Security and Privacy, pp. 926–943 (2018)

    Google Scholar 

  26. Zhang, Y., Genkin, D., Katz, J., Papadopoulos, D., Papamanthou, C.: vSQL: verifying arbitrary SQL queries over dynamic outsourced databases. In: 2017 IEEE Symposium on Security and Privacy, pp. 863–880 (2017)

    Google Scholar 

Download references

Acknowledgements

Mingjie Chen and Christophe Petit are partly supported by EPSRC through grant number EP/V011324/1. Yi-Fu Lai thanks the New Zealand Ministry for Business and Employment for financial support.

Author information

Authors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Mingjie Chen & Christophe Petit

  2. University of Auckland, Auckland, New Zealand

    Yi-Fu Lai

  3. Ruhr-University Bochum, Bochum, Germany

    Yi-Fu Lai

  4. Université Libre de Bruxelles, Bruxelles, Belgium

    Abel Laval & Christophe Petit

  5. EPFL, Lausanne, Switzerland

    Laurane Marco

Authors
  1. Mingjie Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi-Fu Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abel Laval
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Laurane Marco
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Christophe Petit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abel Laval .

Editor information

Editors and Affiliations

  1. Nanyang Technological University, Singapore, Singapore

    Anupam Chattopadhyay

  2. Nanyang Technological University, Singapore, Singapore

    Shivam Bhasin

  3. Radboud University, Nijmegen, The Netherlands

    Stjepan Picek

  4. Indian Institute of Technology Madras, Chennai, India

    Chester Rebeiro

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chen, M., Lai, YF., Laval, A., Marco, L., Petit, C. (2024). Malleable Commitments from Group Actions and Zero-Knowledge Proofs for Circuits Based on Isogenies. In: Chattopadhyay, A., Bhasin, S., Picek, S., Rebeiro, C. (eds) Progress in Cryptology – INDOCRYPT 2023. INDOCRYPT 2023. Lecture Notes in Computer Science, vol 14459. Springer, Cham. https://doi.org/10.1007/978-3-031-56232-7_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-56232-7_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-56231-0

  • Online ISBN: 978-3-031-56232-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics