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Annual International Conference on the Theory and Applications of Cryptographic Techniques

EUROCRYPT 2023: Advances in Cryptology – EUROCRYPT 2023 pp 499–530Cite as

HyperPlonk: Plonk with Linear-Time Prover and High-Degree Custom Gates

Part of the Lecture Notes in Computer Science book series (LNCS,volume 14005)

Abstract

Plonk is a widely used succinct non-interactive proof system that uses univariate polynomial commitments. Plonk is quite flexible: it supports circuits with low-degree “custom” gates as well as circuits with lookup gates (a lookup gate ensures that its input is contained in a predefined table). For large circuits, the bottleneck in generating a Plonk proof is the need for computing a large FFT.

We present HyperPlonk, an adaptation of Plonk to the boolean hypercube, using multilinear polynomial commitments. HyperPlonk retains the flexibility of Plonk but provides several additional benefits. First, it avoids the need for an FFT during proof generation. Second, and more importantly, it supports custom gates of much higher degree than Plonk without harming the running time of the prover. Both of these can dramatically speed up the prover’s running time. Since HyperPlonk relies on multilinear polynomial commitments, we revisit two elegant constructions: one from Orion and one from Virgo. We show how to reduce the Orion opening proof size to less than 10 KB (an almost factor 1000 improvement) and show how to make the Virgo FRI-based opening proof simpler and shorter (This is an extended abstract. The full version is available on EPRINT[22]).

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Notes

  1. 1.

    The constants depend linearly on the degree of the custom gates. These numbers are for simple degree 2 arithmetic circuits.

  2. 2.

    A more general Plonkish arithmetization [54] supports wider tuples, but triples are sufficient here.

  3. 3.

    These are fields where there exists an element that has a smooth order of at least d.

  4. 4.

    Recent breakthrough results have shown that polynomial multiplication is \(O(d\log (d))\) over arbitrary finite fields [35] and there have been efforts toward building practical, fast multiplication algorithms for arbitrary fields [9]. In practice, and especially for low-degree polynomials, using Karatsuba multiplication might be faster.

  5. 5.

    Here we further require \(|\mathbb {F}| \ge 2^{\mu }\) so that \([\textbf{x}]\) never overflow.

  6. 6.

    We can pad zeroes if the actual number is not a power of two.

  7. 7.

    We can pad with dummy states if the number of steps is not a power of two.

References

  1. Aly, A., Ashur, T., Ben-Sasson, E., Dhooghe, S., Szepieniec, A.: Design of symmetric-key primitives for advanced cryptographic protocols. IACR Trans. Symm. Cryptol. 2020(3), 1–45 (2020). https://doi.org/10.13154/tosc.v2020.i3.1-45

  2. Aranha, D.F., Bennedsen, E.M., Campanelli, M., Ganesh, C., Orlandi, C., Takahashi, A.: ECLIPSE: enhanced compiling method for pedersen-committed zkSNARK engines. Cryptology ePrint Archive, Report 2021/934 (2021). https://eprint.iacr.org/2021/934

  3. Arun, A., Ganesh, C., Lokam, S., Mopuri, T., Sridhar, S.: Dew: transparent constant-sized zkSNARKs. Cryptology ePrint Archive, Report 2022/419 (2022). https://eprint.iacr.org/2022/419

  4. Babai, L., Moran, S.: Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes. J. Comput. Syst. Sci. 36(2), 254–276 (1988)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Barbara, M., et al.: Reinforced concrete: fast hash function for zero knowledge proofs and verifiable computation. Cryptology ePrint Archive, Report 2021/1038 (2021). https://eprint.iacr.org/2021/1038

  6. 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

    CrossRef  Google Scholar 

  7. Ben-Sasson, E., Bentov, I., Horesh, Y., Riabzev, M.: Fast reed-solomon interactive oracle proofs of proximity. In: Chatzigiannakis, I., Kaklamanis, C., Marx, D., Sannella, D. (eds.) ICALP 2018. LIPIcs, vol. 107, pp. 14:1–14:17. Schloss Dagstuhl, July 2018. https://doi.org/10.4230/LIPIcs.ICALP.2018.14

  8. Ben-Sasson, E., Bentov, I., Horesh, Y., Riabzev, M.: Scalable zero knowledge with no trusted setup. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 701–732. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_23

    CrossRef  Google Scholar 

  9. Ben-Sasson, E., Carmon, D., Kopparty, S., Levit, D.: Elliptic curve fast fourier transform (ECFFT) part ii: scalable and transparent proofs over all large fields (2022)

    Google Scholar 

  10. 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. LNCS, vol. 11476, pp. 103–128. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17653-2_4

    CrossRef  Google Scholar 

  11. Ben-Sasson, E., Chiesa, A., Spooner, N.: Interactive oracle proofs. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 31–60. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53644-5_2

    CrossRef  Google Scholar 

  12. Ben-Sasson, E., Sudan, M.: Short pcps with polylog query complexity. SIAM J. Comput. 38(2), 551–607 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again. In: Goldwasser, S. (ed.) ITCS 2012, pp. 326–349. ACM, January 2012. https://doi.org/10.1145/2090236.2090263

  14. 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

    CrossRef  Google Scholar 

  15. Bootle, J., Cerulli, A., Ghadafi, E., Groth, J., Hajiabadi, M., Jakobsen, S.K.: Linear-time zero-knowledge proofs for arithmetic circuit satisfiability. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10626, pp. 336–365. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70700-6_12

    CrossRef  Google Scholar 

  16. Bootle, J., Chiesa, A., Groth, J.: Linear-time arguments with sublinear verification from tensor codes. In: Pass, R., Pietrzak, K. (eds.) TCC 2020. LNCS, vol. 12551, pp. 19–46. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64378-2_2

    CrossRef  Google Scholar 

  17. Bootle, J., Chiesa, A., Hu, Y., Orrù, M.: Gemini: elastic SNARKs for diverse environments. In: Dunkelman, O., Dziembowski, S. (eds.) EUROCRYPT 2022, Part II. LNCS, vol. 13276, pp. 427–457. Springer, Heidelberg, May/June 2022. https://doi.org/10.1007/978-3-031-07085-3_15

  18. 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. IEEE Computer Society Press, May 2018. https://doi.org/10.1109/SP.2018.00020

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

    CrossRef  Google Scholar 

  20. Campanelli, M., Faonio, A., Fiore, D., Querol, A., Rodríguez, H.: Lunar: a toolbox for more efficient universal and updatable zkSNARKs and commit-and-prove extensions. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021. LNCS, vol. 13092, pp. 3–33. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92078-4_1

    CrossRef  Google Scholar 

  21. Campanelli, M., Fiore, D., Querol, A.: LegoSNARK: modular design and composition of succinct zero-knowledge proofs. In: Cavallaro, L., Kinder, J., Wang, X., Katz, J. (eds.) ACM CCS 2019, pp. 2075–2092. ACM Press, November 2019. https://doi.org/10.1145/3319535.3339820

  22. Chen, B., Bünz, B., Boneh, D., Zhang, Z.: HyperPlonk: plonk with linear-time prover and high-degree custom gates. Cryptology ePrint Archive, Report 2022/1355 (2022). https://eprint.iacr.org/2022/1355

  23. Chiesa, A., Forbes, M.A., Spooner, N.: A zero knowledge sumcheck and its applications. Cryptology ePrint Archive, Report 2017/305 (2017). https://eprint.iacr.org/2017/305

  24. 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. LNCS, vol. 12105, pp. 738–768. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45721-1_26

    CrossRef  Google Scholar 

  25. Drake, J.: Plonk-style SNARKs without FFTs (2019). https://notes.ethereum.org/DLRqK9V7RIOsTZkab8HmQ?view

  26. Gabizon, A.: Multiset checks in plonk and plookup. https://hackmd.io/@arielg/ByFgSDA7D

  27. Gabizon, A., Williamson, Z.J.: plookup: a simplified polynomial protocol for lookup tables. Cryptology ePrint Archive, Report 2020/315 (2020). https://eprint.iacr.org/2020/315

  28. Gabizon, A., Williamson, Z.J.: Proposal: the turbo-plonk program syntax for specifying snark programs (2020). https://docs.zkproof.org/pages/standards/accepted-workshop3/proposal-turbo_plonk.pdf

  29. 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://eprint.iacr.org/2019/953

  30. 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

    CrossRef  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. Golovnev, A., Lee, J., Setty, S., Thaler, J., Wahby, R.S.: Brakedown: linear-time and post-quantum SNARKs for R1CS. Cryptology ePrint Archive, Report 2021/1043 (2021). https://eprint.iacr.org/2021/1043

  33. Grassi, L., Khovratovich, D., Rechberger, C., Roy, A., Schofnegger, M.: Poseidon: a new hash function for zero-knowledge proof systems. In: Bailey, M., Greenstadt, R. (eds.) USENIX Security 2021, pp. 519–535. USENIX Association, August 2021

    Google Scholar 

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

    CrossRef  Google Scholar 

  35. Harvey, D., Van Der Hoeven, J.: Polynomial multiplication over finite fields in time. J. ACM (JACM) 69(2), 1–40 (2022)

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. Kate, A., Zaverucha, G.M., Goldberg, I.: Constant-size commitments to polynomials and their applications. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 177–194. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17373-8_11

    CrossRef  Google Scholar 

  37. Kattis, A.A., Panarin, K., Vlasov, A.: RedShift: transparent SNARKs from list polynomial commitments. In: Yin, H., Stavrou, A., Cremers, C., Shi, E. (eds.) ACM CCS 2022, pp. 1725–1737. ACM Press, November 2022. https://doi.org/10.1145/3548606.3560657

  38. 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

    CrossRef  Google Scholar 

  39. Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. J. ACM (JACM) 39(4), 859–868 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. Papamanthou, C., Shi, E., Tamassia, R.: Signatures of correct computation. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 222–242. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36594-2_13

    CrossRef  MATH  Google Scholar 

  41. Pearson, L., Fitzgerald, J., Masip, H., Bellés-Muñoz, M., Muñoz-Tapia, J.L.: PlonKup: reconciling PlonK with plookup. Cryptology ePrint Archive, Report 2022/086 (2022). https://eprint.iacr.org/2022/086

  42. Posen, J., Kattis, A.A.: Caulk+: table-independent lookup arguments. Cryptology ePrint Archive, Report 2022/957 (2022). https://eprint.iacr.org/2022/957

  43. Setty, S.: Spartan: efficient and general-purpose zkSNARKs without trusted setup. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12172, pp. 704–737. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56877-1_25

    CrossRef  Google Scholar 

  44. Setty, S., Lee, J.: Quarks: quadruple-efficient transparent zkSNARKs. Cryptology ePrint Archive, Report 2020/1275 (2020). https://eprint.iacr.org/2020/1275

  45. System, E.: Jellyfish jellyfish cryptographic library (2022). https://github.com/EspressoSystems/jellyfish

  46. Thaler, J.: Time-optimal interactive proofs for circuit evaluation. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 71–89. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_5

    CrossRef  Google Scholar 

  47. Thaler, J.: Proofs, arguments, and zero-knowledge (2020)

    Google Scholar 

  48. 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. IEEE Computer Society Press, May 2018. https://doi.org/10.1109/SP.2018.00060

  49. Xie, T., Zhang, J., Zhang, Y., Papamanthou, C., Song, D.: Libra: succinct zero-knowledge proofs with optimal prover computation. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 733–764. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_24

    CrossRef  Google Scholar 

  50. Xie, T., Zhang, Y., Song, D.: Orion: zero knowledge proof with linear prover time. Cryptology ePrint Archive, Report 2022/1010 (2022). https://eprint.iacr.org/2022/1010

  51. Xie, T., Zhang, Y., Song, D.: Orion: zero knowledge proof with linear prover time. In: Dodis, Y., Shrimpton, T. (eds.) CRYPTO 2022, Part IV. LNCS, vol. 13510, pp. 299–328. Springer, Heidelberg, August 2022. https://doi.org/10.1007/978-3-031-15985-5_11

  52. Xiong, A.L., et al.: VERI-ZEXE: decentralized private computation with universal setup. Cryptology ePrint Archive, Report 2022/802 (2022). https://eprint.iacr.org/2022/802

  53. Zapico, A., Buterin, V., Khovratovich, D., Maller, M., Nitulescu, A., Simkin, M.: Caulk: lookup arguments in sublinear time. In: Yin, H., Stavrou, A., Cremers, C., Shi, E. (eds.) ACM CCS 2022, pp. 3121–3134. ACM Press, November 2022. https://doi.org/10.1145/3548606.3560646

  54. Zcash: PLONKish arithmetization. https://zcash.github.io/halo2/concepts/arithmetization.html (2022)

  55. Zhang, J., Xie, T., Zhang, Y., Song, D.: Transparent polynomial delegation and its applications to zero knowledge proof. In: 2020 IEEE Symposium on Security and Privacy, pp. 859–876. IEEE Computer Society Press, May 2020. https://doi.org/10.1109/SP40000.2020.00052

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Authors and Affiliations

  1. Espresso Systems, Menlo Park, USA

    Binyi Chen, Benedikt Bünz & Zhenfei Zhang

  2. Stanford University, Stanford, USA

    Benedikt Bünz & Dan Boneh

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  1. Binyi Chen
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  1. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay

  2. Simula UiB, Bergen, Norway

    Martijn Stam

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Chen, B., Bünz, B., Boneh, D., Zhang, Z. (2023). HyperPlonk: Plonk with Linear-Time Prover and High-Degree Custom Gates. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14005. Springer, Cham. https://doi.org/10.1007/978-3-031-30617-4_17

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