Skip to main content
SpringerLink
Log in

SNARGs for P from Sub-exponential DDH and QR

  • Conference paper
  • First Online:
Advances in Cryptology – EUROCRYPT 2022 (EUROCRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13276))

Abstract

We obtain publicly verifiable Succinct Non-Interactive Arguments (SNARGs) for arbitrary deterministic computations and bounded space non-deterministic computation from standard group-based assumptions, without relying on pairings. In particular, assuming the sub-exponential hardness of both the Decisional Diffie-Hellman (DDH) and Quadratic Residuosity (QR) assumptions, we obtain the following results, where n denotes the length of the instance:

  1. 1.

    A SNARG for any language that can be decided in non-deterministic time T and space S with communication complexity and verifier runtime \((n+S) \cdot T^{o(1)}\).

  2. 2.

    A SNARG for any language that can be decided in deterministic time T with communication complexity and verifier runtime \(n \cdot T^{o(1)}\).

J. Hulett, R. Jawale and D. Khurana—Supported in part by DARPA SIEVE award under contract number HR001120C0024, a gift from Visa Research, and a C3AI DTI award. 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 United States Government or DARPA.

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 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
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.

    Batch arguments for \(\mathsf {NP}\) allow a verifier to verify the correctness of k \(\mathsf {NP}\) instances with circuit complexity smaller than k times the size of the \(\mathsf {NP}\) verification circuit.

  2. 2.

    In the full version of this paper, we show that one can in fact construct a commitment with somewhat succinct local openings from DDH or QR. However, these are significantly less succinct than their LWE-based counterparts, and using these commitments would lead to marginally worse parameters than one can get with the methods described next.

  3. 3.

    We remark that [20] also require some additional linear homomorphism properties from the commitment, but these are not necessary for our discussion.

  4. 4.

    This becomes somewhat non-trivial in the non-deterministic setting, which we discuss in an upcoming subsection.

  5. 5.

    For simplicity of exposition, we are here ignoring some additional additive overhead as well as polylogarithmic multiplicative factors.

  6. 6.

    More generally, this can be computed as a function of the entire transcript.

  7. 7.

    If a non-deterministic Turing Machine wishes to remember what non-deterministic choices it made, it has to write them down to its work tape.

References

  1. Ananth, P., Chen, Y.-C., Chung, K.-M., Lin, H., Lin, W.-K.: Delegating RAM computations with adaptive soundness and privacy. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 3–30. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53644-5_1

    Chapter  Google Scholar 

  2. Badrinarayanan, S., Fernando, R., Jain, A., Khurana, D., Sahai, A.: Statistical ZAP arguments. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12107, pp. 642–667. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_22

    Chapter  Google Scholar 

  3. Badrinarayanan, S., Kalai, Y.T., Khurana, D., Sahai, A., Wichs, D.: Succinct delegation for low-space non-deterministic computation. In: STOC, pp. 709–721 (2018)

    Google Scholar 

  4. Barak, B.: How to go beyond the black-box simulation barrier. In: FOCS, pp. 106–115 (2001)

    Google Scholar 

  5. Bartusek, J., Bronfman, L., Holmgren, J., Ma, F., Rothblum, R.D.: On the (in)security of Kilian-based SNARGs. In: Hofheinz, D., Rosen, A. (eds.) TCC 2019. LNCS, vol. 11892, pp. 522–551. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-36033-7_20

    Chapter  MATH  Google Scholar 

  6. Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: Denning, D.E., Pyle, R., Ganesan, R., Sandhu, R.S., Ashby, V. (eds.) ACM Conference on Computer and Communications Security, pp. 62–73. ACM (1993)

    Google Scholar 

  7. Bitansky, N., et al.: The hunting of the SNARK. IACR Cryptol. ePrint Arch. 2014, 580 (2014). http://eprint.iacr.org/2014/580

  8. Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: Recursive composition and bootstrapping for SNARKS and proof-carrying data. In: STOC, pp. 111–120 (2013)

    Google Scholar 

  9. 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, http://dx.doi.org/10.1007/978-3-642-36594-2_18

  10. Bitansky, N., Garg, S., Lin, H., Pass, R., Telang, S.: Succinct randomized encodings and their applications. IACR Cryptology ePrint Archive 2015, 356 (2015)

    Google Scholar 

  11. Bitansky, N., Kalai, Y.T., Paneth, O.: Multi-collision resistance: a paradigm for keyless hash functions. In: Diakonikolas, I., Kempe, D., Henzinger, M. (eds.) STOC, pp. 671–684. ACM (2018)

    Google Scholar 

  12. Brakerski, Z., Holmgren, J., Kalai, Y.T.: Non-interactive delegation and batch NP verification from standard computational assumptions. In: STOC, pp. 474–482 (2017)

    Google Scholar 

  13. Brakerski, Z., Kalai, Y.: Witness indistinguishability for any single-round argument with applications to access control. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020. LNCS, vol. 12111, pp. 97–123. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45388-6_4

    Chapter  Google Scholar 

  14. Brakerski, Z., Koppula, V., Mour, T.: NIZK from LPN and trapdoor hash via correlation intractability for approximable relations. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12172, pp. 738–767. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56877-1_26

    Chapter  Google Scholar 

  15. Canetti, R., et al.: Fiat-Shamir: from practice to theory. In: Charikar, M., Cohen, E. (eds.) STOC, pp. 1082–1090. ACM (2019)

    Google Scholar 

  16. Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited. J. ACM 51(4), 557–594 (2004)

    Article  MathSciNet  Google Scholar 

  17. Canetti, R., Holmgren, J.: Fully succinct garbled RAM. In: ITCS, pp. 169–178. ACM (2016)

    Google Scholar 

  18. Canetti, R., Holmgren, J., Jain, A., Vaikuntanathan, V.: Succinct garbling and indistinguishability obfuscation for RAM programs. In: STOC, pp. 429–437. ACM (2015)

    Google Scholar 

  19. Chen, Y., Chow, S.S.M., Chung, K., Lai, R.W.F., Lin, W., Zhou, H.: Cryptography for parallel RAM from indistinguishability obfuscation. In: ITCS, pp. 179–190. ACM (2016)

    Google Scholar 

  20. Choudhuri, A.R., Jain, A., Jin, Z.: Non-interactive batch arguments for NP from standard assumptions. IACR Cryptol. ePrint Arch. 2021, 807 (2021). https://eprint.iacr.org/2021/807

  21. Choudhuri, A.R., Jain, A., Jin, Z.: Snargs for P from LWE. IACR Cryptol. ePrint Arch, p. 808 (2021). https://eprint.iacr.org/2021/808

  22. Damgård, I., Faust, S., Hazay, C.: Secure two-party computation with low communication. In: Theory of Cryptography–9th Theory of Cryptography Conference, TCC 2012, Taormina, Sicily, Italy, 19–21 March 2012. Proceedings, pp. 54–74 (2012). https://doi.org/10.1007/978-3-642-28914-9_4, http://dx.doi.org/10.1007/978-3-642-28914-9_4

  23. 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, http://dx.doi.org/10.1007/978-3-642-38348-9_37

  24. Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: STOC, pp. 99–108 (2011)

    Google Scholar 

  25. Goldwasser, S., Kalai, Y.T.: On the (in)security of the fiat-shamir paradigm. In: FOCS, p. 102 (2003)

    Google Scholar 

  26. Goldwasser, S., Kalai, Y.T., Rothblum, G.N.: Delegating computation: interactive proofs for muggles. J. ACM 62(4), 27 (2015)

    Article  MathSciNet  Google Scholar 

  27. González, A., Zacharakis, A.: Fully-succinct publicly verifiable delegation from constant-size assumptions. In: Nissim, K., Waters, B. (eds.) TCC 2021. LNCS, vol. 13042, pp. 529–557. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90459-3_18

    Chapter  Google Scholar 

  28. Goyal, V., Jain, A., Jin, Z., Malavolta, G.: Statistical zaps and new oblivious transfer protocols. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12107, pp. 668–699. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_23

    Chapter  Google Scholar 

  29. 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 

  30. Hubácek, P., Wichs, D.: On the communication complexity of secure function evaluation with long output. In: Roughgarden, T. (ed.) Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, 11–13 January 2015, pp. 163–172. ACM (2015). https://doi.org/10.1145/2688073.2688105

  31. Jain, A., Jin, Z.: Non-interactive zero knowledge from sub-exponential DDH. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12696, pp. 3–32. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_1

    Chapter  Google Scholar 

  32. Jawale, R., Kalai, Y.T., Khurana, D., Zhang, R.: SNARGs for bounded depth computations and PPAD hardness from sub-exponential LWE. In: Khuller, S., Williams, V.V. (eds.) STOC 2021: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, 21–25 June 2021, pp. 708–721. ACM (2021). https://doi.org/10.1145/3406325.3451055

  33. Kalai, Y.T., Paneth, O.: Delegating RAM computations. In: Theory of Cryptography - 14th International Conference, TCC 2016-B, Beijing, China, 31 October–3 November 2016, Proceedings, Part II, pp. 91–118 (2016). https://doi.org/10.1007/978-3-662-53644-5_4

  34. Kalai, Y.T., Paneth, O., Yang, L.: How to delegate computations publicly. 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. 1115–1124. ACM (2019). https://doi.org/10.1145/3313276.3316411

  35. Kalai, Y.T., Raz, R., Rothblum, R.D.: Delegation for bounded space. In: Symposium on Theory of Computing Conference, STOC 2013, Palo Alto, CA, USA, 1–4 June 2013, pp. 565–574 (2013). https://doi.org/10.1145/2488608.2488679

  36. Kalai, Y.T., Raz, R., Rothblum, R.D.: How to delegate computations: the power of no-signaling proofs. In: STOC, pp. 485–494. ACM (2014)

    Google Scholar 

  37. Kalai, Y.T., Vaikuntanathan, V., Zhang, R.Y.: Somewhere statistical soundness, post-quantum security, and snargs. Cryptology ePrint Archive, Report 2021/788 (2021). https://ia.cr/2021/788

  38. Kilian, J.: A note on efficient zero-knowledge proofs and arguments (extended abstract). In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pp. 723–732. ACM (1992)

    Google Scholar 

  39. Koppula, V., Lewko, A.B., Waters, B.: Indistinguishability obfuscation for turing machines with unbounded memory. In: STOC, pp. 419–428. ACM (2015)

    Google Scholar 

  40. 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 

  41. Lombardi, A., Vaikuntanathan, V., Wichs, D.: Statistical ZAPR arguments from bilinear maps. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12107, pp. 620–641. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_21

    Chapter  Google Scholar 

  42. Micali, S.: CS proofs (extended abstracts). In: 35th Annual Symposium on Foundations of Computer Science, Santa Fe, New Mexico, USA, 20–22 November 1994, pp. 436–453 (1994). full version in [?]. https://doi.org/10.1109/SFCS.1994.365746, http://dx.doi.org/10.1109/SFCS.1994.365746

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

    Chapter  Google Scholar 

  44. Parno, B., Raykova, M., Vaikuntanathan, V.: How to delegate and verify in public: verifiable computation from attribute-based encryption. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 422–439. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28914-9_24

    Chapter  Google Scholar 

  45. 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

    Chapter  Google Scholar 

  46. Pointcheval, D., Stern, J.: Security proofs for signature schemes. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 387–398. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68339-9_33

    Chapter  Google Scholar 

  47. Reingold, O., Rothblum, G.N., Rothblum, R.D.: Constant-round interactive proofs for delegating computation. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, 18–21 June 2016, pp. 49–62 (2016). https://doi.org/10.1145/2897518.2897652, http://doi.acm.org/10.1145/2897518.2897652

Download references

Author information

Authors and Affiliations

  1. University of Illinois, Urbana-Champaign, USA

    James Hulett, Ruta Jawale, Dakshita Khurana & Akshayaram Srinivasan

  2. Tata Institute of Fundamental Research, Bengaluru, India

    James Hulett, Ruta Jawale, Dakshita Khurana & Akshayaram Srinivasan

Authors
  1. James Hulett
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruta Jawale
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dakshita Khurana
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Akshayaram Srinivasan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dakshita Khurana .

Editor information

Editors and Affiliations

  1. University of Haifa, Haifa, Haifa, Israel

    Orr Dunkelman

  2. University of Warsaw, Warsaw, Poland

    Stefan Dziembowski

Rights and permissions

Reprints and permissions

Copyright information

© 2022 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Hulett, J., Jawale, R., Khurana, D., Srinivasan, A. (2022). SNARGs for P from Sub-exponential DDH and QR. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13276. Springer, Cham. https://doi.org/10.1007/978-3-031-07085-3_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-07085-3_18

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-07084-6

  • Online ISBN: 978-3-031-07085-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation