Skip to main content
SpringerLink
Log in

Shorter Pairing-Based Arguments Under Standard Assumptions

  • Conference paper
  • First Online:
Advances in Cryptology – ASIACRYPT 2019 (ASIACRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11923))

Abstract

This paper constructs efficient non-interactive arguments for correct evaluation of arithmetic and boolean circuits with proof size O(d) group elements, where d is the multiplicative depth of the circuit, under falsifiable assumptions. This is achieved by combining techniques from SNARKs and QA-NIZK arguments of membership in linear spaces. The first construction is very efficient (the proof size is \(\approx 4d\) group elements and the verification cost is \(\approx 4d\) pairings and \(O(n+n'+d)\) exponentiations, where n is the size of the input and \(n'\) of the output) but one type of attack can only be ruled out assuming the knowledge soundness of QA-NIZK arguments of membership in linear spaces. We give an alternative construction which replaces this assumption with a decisional assumption in bilinear groups at the cost of approximately doubling the proof size. The construction for boolean circuits can be made zero-knowledge with Groth-Sahai proofs, resulting in a NIZK argument for circuit satisfiability based on falsifiable assumptions in bilinear groups of proof size \(O(n+d)\).

Our main technical tool is what we call an “argument of knowledge transfer”. Given a commitment \(C_1\) and an opening x, such an argument allows to prove that some other commitment \(C_2\) opens to f(x), for some function f, even if \(C_2\) is not extractable. We construct very short, constant-size, pairing-based arguments of knowledge transfer with constant-time verification for any linear function and also for Hadamard products. These allow to transfer the knowledge of the input to lower levels of the circuit.

A. González—This author was supported in part by the French ANR ALAMBIC project (ANR-16-CE39-0006).

C. Ràfols—The research leading to this article was supported by a Marie Curie “UPF Fellows” Postdoctoral Grant and by Project RTI2018-102112-B-I00 (AEI/FEDER, UE).

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

Notes

  1. 1.

    Essentially the only such commitment known is bit to bit encryption, e.g. Groth-Sahai commitments to bits.

  2. 2.

    Note that using the celebrated recent results of Peikert and Shiehian [35] this scheme can be based solely on the LWE assumption.

  3. 3.

    There’s the technicality that a verifier running in time sub-linear in the circuit size can not even read the circuit, which is part of the input of the universal circuit. For this reason, they restricted the circuits to be log space uniform boolean cicuits.

  4. 4.

    For the distribution \(\mathcal {M}^{\top }\) used in Sect. 5 this assumption is equivalent to the m-DLog assumption.

  5. 5.

    This part of the proof follows essentially the same lines of the first constant-size QA-NIZK arguments for linear spaces of Libert et al. [26] which were later simplified and generalized by Kiltz and Wee [25].

  6. 6.

    We can assume w.l.o.g. that the crs for the linear knowledge transfer associated to \(\mathbf {{M}}_i,\mathbf {{N}}_i,\mathbf {{P}}_i\) includes \(\{[s_j]_{1,2}\}_{j=1}^{N-1},[s_j^N]\), as this does not compromise security.

References

  1. Danezis, G., Fournet, C., Groth, J., Kohlweiss, M.: Square span programs with applications to succinct NIZK arguments. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014, Part I. LNCS, vol. 8873, pp. 532–550. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45611-8_28

    Chapter  Google Scholar 

  2. Daza, V., González, A., Pindado, Z., Ràfols, C., Silva, J.: Shorter quadratic QA-NIZK proofs. In: Lin, D., Sako, K. (eds.) PKC 2019, Part I. LNCS, vol. 11442, pp. 314–343. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17253-4_11

    Chapter  Google Scholar 

  3. Escala, A., Groth, J.: Fine-tuning Groth-Sahai proofs. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 630–649. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54631-0_36

    Chapter  Google Scholar 

  4. Escala, A., Herold, G., Kiltz, E., Ràfols, C., Villar, J.L.: An algebraic framework for Diffie-Hellman assumptions. J. Cryptol. 30(1), 242–288 (2017)

    Article  MathSciNet  Google Scholar 

  5. Fauzi, P., Lipmaa, H., Siim, J., Zając, M.: An efficient pairing-based shuffle argument. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017, Part II. LNCS, vol. 10625, pp. 97–127. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70697-9_4

    Chapter  Google Scholar 

  6. Ferrara, A.L., Green, M., Hohenberger, S., Pedersen, M.Ø.: Practical short signature batch verification. In: Fischlin, M. (ed.) CT-RSA 2009. LNCS, vol. 5473, pp. 309–324. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00862-7_21

    Chapter  Google Scholar 

  7. Gennaro, R., Gentry, C., Parno, B.: Non-interactive verifiable computing: outsourcing computation to untrusted workers. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 465–482. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_25

    Chapter  Google Scholar 

  8. 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 

  9. Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) 41st ACM STOC, pp. 169–178. ACM Press, May/June (2009)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: Fortnow, L., Vadhan, S.P. (eds.) 43rd ACM STOC, pp. 99–108. ACM Press, June 2011

    Google Scholar 

  12. Ghadafi, E., Groth, J.: Towards a classification of non-interactive computational assumptions in cyclic groups. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017, Part II. LNCS, vol. 10625, pp. 66–96. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70697-9_3

    Chapter  MATH  Google Scholar 

  13. Goldwasser, S., Kalai, Y.T., Rothblum, G.N.: Delegating computation: interactive proofs for muggles. In: Ladner, R.E., Dwork, C. (eds.) 40th ACM STOC, pp. 113–122. ACM Press, May 2008

    Google Scholar 

  14. González, A., Hevia, A., Ràfols, C.: QA-NIZK arguments in asymmetric groups: new tools and new constructions. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015, Part I. LNCS, vol. 9452, pp. 605–629. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48797-6_25

    Chapter  MATH  Google Scholar 

  15. 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 

  16. 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 

  17. Groth, J., Ostrovsky, R., Sahai, A.: New techniques for noninteractive zero-knowledge. J. ACM 59(3), 11 (2012)

    Article  MathSciNet  Google Scholar 

  18. Groth, J., Sahai, A.: Efficient noninteractive proof systems for bilinear groups. SIAM J. Comput. 41(5), 1193–1232 (2012)

    Article  MathSciNet  Google Scholar 

  19. Herold, G., Hoffmann, M., Klooß, M., Ràfols, C., Rupp, A.: New techniques for structural batch verification in bilinear groups with applications to groth-sahai proofs. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D. (eds.) ACM CCS 2017, pp. 1547–1564. ACM Press, October/November (2017)

    Google Scholar 

  20. Jutla, C.S., Roy, A.: Shorter quasi-adaptive NIZK proofs for linear subspaces. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part I. LNCS, vol. 8269, pp. 1–20. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42033-7_1

    Chapter  Google Scholar 

  21. Jutla, C.S., Roy, A.: Switching lemma for bilinear tests and constant-size NIZK proofs for linear subspaces. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part II. LNCS, vol. 8617, pp. 295–312. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44381-1_17

    Chapter  Google Scholar 

  22. Kalai, Y., Paneth, O., Yang, L.: On publicly verifiable delegation from standard assumptions. Cryptology ePrint Archive, Report 2018/776 (2018). https://eprint.iacr.org/2018/776

  23. Kalai, Y.T., Raz, R., Rothblum, R.D.: How to delegate computations: the power of no-signaling proofs. In: Shmoys, D.B. (ed.) 46th ACM STOC, pp. 485–494. ACM Press, May/June (2014)

    Google Scholar 

  24. Katsumata, S., Nishimaki, R., Yamada, S., Yamakawa, T.: Exploring constructions of compact NIZKs from various assumptions. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. Part III, volume 11694 of LNCS, pp. 639–669. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-030-26954-8_21

    Chapter  Google Scholar 

  25. Kiltz, E., Wee, H.: Quasi-adaptive NIZK for linear subspaces revisited. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015, Part II. LNCS, vol. 9057, pp. 101–128. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_4

    Chapter  Google Scholar 

  26. Libert, B., Peters, T., Joye, M., Yung, M.: Non-malleability from malleability: simulation-sound quasi-adaptive NIZK proofs and CCA2-secure encryption from homomorphic signatures. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 514–532. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_29

    Chapter  Google Scholar 

  27. Libert, B., Peters, T., Yung, M.: Short group signatures via structure-preserving signatures: standard model security from simple assumptions. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015, Part II. LNCS, vol. 9216, pp. 296–316. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_15

    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. Lund, C., Fortnow, L., Karloff, H.J., Nisan, N.: Algebraic methods for interactive proof systems. In: 31st FOCS, pp. 2–10. IEEE Computer Society Press, October 1990

    Google Scholar 

  30. Maller, M., Kohlweiss, M., Bowe, S., Meiklejohn, S.: Sonic: zero-knowledge snarks from linear-size universal and updateable structured reference string. Cryptology ePrint Archive, Report 2019/099 (2019). http://eprint.iacr.org/2019/099

  31. Morillo, P., Ràfols, C., Villar, J.L.: The kernel matrix diffie-hellman assumption. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016, Part I. LNCS, vol. 10031, pp. 729–758. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_27

    Chapter  Google Scholar 

  32. Naor, M.: On cryptographic assumptions and challenges. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 96–109. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_6

    Chapter  Google Scholar 

  33. Paneth, O., Rothblum, G.N.: On Zero-testable homomorphic encryption and publicly verifiable non-interactive arguments. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part II. LNCS, vol. 10678, pp. 283–315. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70503-3_9

    Chapter  Google Scholar 

  34. 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 

  35. Peikert, C., Shiehian, S.: Noninteractive zero knowledge for NP from (plain) learning with errors. Cryptology ePrint Archive, Report 2019/158 (2019). https://eprint.iacr.org/2019/158

Download references

Author information

Authors and Affiliations

  1. ENS de Lyon, Laboratoire LIP (U. Lyon, CNRS, ENSL, INRIA, UCBL), Lyon, France

    Alonso González

  2. Universitat Pompeu Fabra and Cybercat, Barcelona, Spain

    Carla Ràfols

Authors
  1. Alonso González
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carla Ràfols
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alonso González .

Editor information

Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

Rights and permissions

Reprints and permissions

Copyright information

© 2019 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

González, A., Ràfols, C. (2019). Shorter Pairing-Based Arguments Under Standard Assumptions. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11923. Springer, Cham. https://doi.org/10.1007/978-3-030-34618-8_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-34618-8_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-34617-1

  • Online ISBN: 978-3-030-34618-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation