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Private Polynomial Commitments and Applications to MPC

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13941))

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Abstract

Polynomial commitment schemes allow a prover to commit to a polynomial and later reveal the evaluation of the polynomial on an arbitrary point along with proof of validity. This object is central in the design of many cryptographic schemes such as zero-knowledge proofs and verifiable secret sharing. In the standard definition, the polynomial is known to the prover whereas the evaluation points are not private. In this paper, we put forward the notion of private polynomial commitments that capture additional privacy guarantees, where the evaluation points are hidden from the verifier while the polynomial is hidden from both.

We provide concretely efficient constructions that allow simultaneously batch the verification of many evaluations with a small additive overhead. As an application, we design a new concretely efficient multi-party private set-intersection with malicious security and improved asymptotic communication and space complexities.

We demonstrate the concrete efficiency of our construction via an implementation. Our scheme can prove \(2^{10}\) evaluations of a private polynomial of degree \(2^{10}\) in 157 s. The proof size is only 169 KB and the verification time is 11.8 s. Moreover, we also implemented the multi-party private set intersection protocol and scale it to 1000 parties (which has not been shown before). The total running time for \(2^{14}\) elements per party is 2,410 s. While existing protocols offer better computational complexity, our scheme offers significantly smaller communication and better scalability (in the number of parties) owing to better memory usage.

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References

  1. Ate pairing. https://github.com/herumi/ate-pairing

  2. The GNU multiple precision arithmetic library. https://gmplib.org/

  3. Abascal, J., Sereshgi, M.H.F., Hazay, C., Ishai, Y., Venkitasubramaniam, M.: Is the classical GMW paradigm practical? the case of non-interactive actively secure 2pc. In: CCS, pp. 1591–1605 (2020)

    Google Scholar 

  4. Abe, M., Fuchsbauer, G., Groth, J., Haralambiev, K., Ohkubo, M.: Structure-preserving signatures and commitments to group elements. J. Cryptol. 29, 363–421 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Afshar, A., Mohassel, P., Pinkas, B., Riva, B.: Non-interactive secure computation based on cut-and-choose. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 387–404. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_22

    Chapter  Google Scholar 

  6. Backes, M., Datta, A., Kate, A.: Asynchronous computational VSS with reduced communication complexity. In: CT-RSA, vol. 7779, pp. 259–276 (2013)

    Google Scholar 

  7. Backes, M., Fiore, D., Reischuk, R.M.: Verifiable delegation of computation on outsourced data. In: CCS, pp. 863–874 (2013)

    Google Scholar 

  8. Bayer, S., Groth, J.: Zero-knowledge argument for polynomial evaluation with application to blacklists. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 646–663. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_38

    Chapter  Google Scholar 

  9. Ben-Efraim, A., Nissenbaum, O., Omri, E., Paskin-Cherniavsky, A.: Psimple: practical multiparty maliciously-secure private set intersection. In: ASIA CCS, pp. 1098–1112 (2022)

    Google Scholar 

  10. Benabbas, S., Gennaro, R., Vahlis, Y.: Verifiable delegation of computation over large datasets. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 111–131. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_7

    Chapter  Google Scholar 

  11. Bhadauria, R., Hazay, C.: Multi-clients verifiable computation via conditional disclosure of secrets. In: SCN, pp. 150–171 (2020)

    Google Scholar 

  12. Bois, A., Cascudo, I., Fiore, D., Kim, D.: Flexible and efficient verifiable computation on encrypted data. In: Garay, J.A. (ed.) PKC 2021. LNCS, vol. 12711, pp. 528–558. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75248-4_19

    Chapter  Google Scholar 

  13. Boneh, D., Boyen, X., Shacham, H.: Short group signatures. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 41–55. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-28628-8_3

    Chapter  Google Scholar 

  14. Bünz, B., Bootle, J., Boneh, D., Poelstra, A., Wuille, P., Maxwell, G.: Bulletproofs: short proofs for confidential transactions and more. In: IEEE S &P, pp. 315–334 (2018)

    Google Scholar 

  15. 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

    Chapter  Google Scholar 

  16. Bünz, B., Maller, M., Mishra, P., Tyagi, N., Vesely, P.: Proofs for inner pairing products and applications. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021. LNCS, vol. 13092, pp. 65–97. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92078-4_3

    Chapter  Google Scholar 

  17. Camenisch, J., Dubovitskaya, M., Haralambiev, K., Kohlweiss, M.: Composable and modular anonymous credentials: definitions and practical constructions. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9453, pp. 262–288. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48800-3_11

    Chapter  Google Scholar 

  18. Catalano, D., Fiore, D.: Vector commitments and their applications. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 55–72. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36362-7_5

    Chapter  Google Scholar 

  19. Catalano, D., Fiore, D., Gennaro, R., Vamvourellis, K.: Algebraic (Trapdoor) one-way functions and their applications. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 680–699. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36594-2_38

    Chapter  Google Scholar 

  20. Catalano, D., Fiore, D., Warinschi, B.: Homomorphic signatures with efficient verification for polynomial functions. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 371–389. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_21

    Chapter  Google Scholar 

  21. Chase, M., Miao, P.: Private set intersection in the internet setting from lightweight oblivious PRF. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12172, pp. 34–63. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56877-1_2

    Chapter  Google Scholar 

  22. Chepurnoy, A., Papamanthou, C., Zhang, Y.: Edrax: a cryptocurrency with stateless transaction validation. IACR Cryptol. ePrint Arch., p. 968 (2018)

    Google Scholar 

  23. 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

    Chapter  Google Scholar 

  24. Choi, S.G., Katz, J., Kumaresan, R., Cid, C.: Multi-client non-interactive verifiable computation. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 499–518. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36594-2_28

    Chapter  Google Scholar 

  25. De Cristofaro, E., Kim, J., Tsudik, G.: Linear-complexity private set intersection protocols secure in malicious model. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 213–231. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17373-8_13

    Chapter  MATH  Google Scholar 

  26. Fenske, E., Mani, A., Johnson, A., Sherr, M.: Distributed measurement with private set-union cardinality. In: CCS, pp. 2295–2312 (2017)

    Google Scholar 

  27. Fiore, D., Gennaro, R.: Publicly verifiable delegation of large polynomials and matrix computations, with applications. In: CCS, pp. 501–512 (2012)

    Google Scholar 

  28. Fiore, D., Gennaro, R., Pastro, V.: Efficiently encrypted data. In: ACM SIGSAC, pp. 844–855 (2014)

    Google Scholar 

  29. Fiore, D., Nitulescu, A., Pointcheval, D.: Boosting verifiable computation on encrypted data. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020. LNCS, vol. 12111, pp. 124–154. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45388-6_5

    Chapter  Google Scholar 

  30. Freedman, M.J., Ishai, Y., Pinkas, B., Reingold, O.: Keyword search and oblivious pseudorandom functions. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 303–324. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30576-7_17

    Chapter  Google Scholar 

  31. Freedman, M.J., Nissim, K., Pinkas, B.: Efficient private matching and set intersection. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 1–19. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24676-3_1

    Chapter  Google Scholar 

  32. Gabizon, A., Williamson, Z.J., Ciobotaru, O.: Plonk: permutations over lagrange-bases for oecumenical noninteractive arguments of knowledge. IACR Cryptol. ePrint Arch. 2019, 953 (2019)

    Google Scholar 

  33. Garimella, G., Pinkas, B., Rosulek, M., Trieu, N., Yanai, A.: Oblivious key-value stores and amplification for private set intersection. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12826, pp. 395–425. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84245-1_14

    Chapter  Google Scholar 

  34. 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 

  35. Ghosh, S., Nielsen, J.B., Nilges, T.: Maliciously secure oblivious linear function evaluation with constant overhead. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 629–659. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_22

    Chapter  Google Scholar 

  36. Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: How to run turing machines on encrypted data. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 536–553. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_30

    Chapter  Google Scholar 

  37. Gorbunov, S., Reyzin, L., Wee, H., Zhang, Z.: Pointproofs: aggregating proofs for multiple vector commitments. In: ACM SIGSAC, pp. 2007–2023 (2020)

    Google Scholar 

  38. Gordon, S.D., Hazay, C., Le, P.H.: Fully secure PSI via mpc-in-the-head. PoPETS 2022(3), 291–313 (2022)

    Article  Google Scholar 

  39. Gordon, S.D., Katz, J., Liu, F.-H., Shi, E., Zhou, H.-S.: Multi-Client verifiable computation with stronger security guarantees. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 144–168. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_6

    Chapter  Google Scholar 

  40. Hazay, C.: Oblivious polynomial evaluation and secure set-intersection from algebraic PRFs. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 90–120. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_4

    Chapter  Google Scholar 

  41. Hazay, C., Ishai, Y., Venkitasubramaniam, M.: Actively secure garbled circuits with constant communication overhead in the plain model. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10678, pp. 3–39. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70503-3_1

    Chapter  Google Scholar 

  42. Hazay, C., Lindell, Y.: Efficient oblivious polynomial evaluation with simulation-based security. IACR Cryptol. ePrint Arch., p. 459 (2009)

    Google Scholar 

  43. Hazay, C., Venkitasubramaniam, M.: Scalable multi-party private set-intersection. In: Fehr, S. (ed.) PKC 2017. LNCS, vol. 10174, pp. 175–203. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54365-8_8

    Chapter  Google Scholar 

  44. Ishai, Y., Kushilevitz, E., Ostrovsky, R., Prabhakaran, M., Sahai, A.: Efficient non-interactive secure computation. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 406–425. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_23

    Chapter  Google Scholar 

  45. Juels, A., Jr., B.S.K.: Pors: proofs of retrievability for large files. In: CCS, pp. 584–597 (2007)

    Google Scholar 

  46. 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

    Chapter  Google Scholar 

  47. Lee, J.: Dory: efficient, transparent arguments for generalised inner products and polynomial commitments. IACR Cryptol. ePrint Arch. 2020, 1274 (2020)

    Google Scholar 

  48. Mohassel, P., Rosulek, M.: Non-interactive secure 2PC in the offline/online and batch settings. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 425–455. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56617-7_15

    Chapter  MATH  Google Scholar 

  49. Naor, M., Pinkas, B.: Oblivious polynomial evaluation. SIAM J. Comput. 35, 1254–1281 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Nguyen, D.T., Trieu, N.: Mpccache: privacy-preserving multi-party cooperative cache sharing at the edge. IACR Cryptol. ePrint Arch. (2021). https://eprint.iacr.org/2021/317

  51. 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

    Chapter  MATH  Google Scholar 

  52. 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 

  53. Pinkas, B., Rosulek, M., Trieu, N., Yanai, A.: SpOT-light: lightweight private set intersection from sparse OT extension. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 401–431. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_13

    Chapter  Google Scholar 

  54. Pinkas, B., Rosulek, M., Trieu, N., Yanai, A.: PSI from PaXoS: fast, malicious private set intersection. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12106, pp. 739–767. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45724-2_25

    Chapter  Google Scholar 

  55. Pinkas, B., Schneider, T., Tkachenko, O., Yanai, A.: Efficient circuit-based PSI with linear communication. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11478, pp. 122–153. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17659-4_5

    Chapter  Google Scholar 

  56. Raab, M., Steger, A.: “balls into bins" - a simple and tight analysis. In: Randomization and Approximation Techniques in Computer Science, pp. 159–170 (1998)

    Google Scholar 

  57. Rosulek, M., Trieu, N.: Compact and malicious private set intersection for small sets. IACR Cryptol. ePrint Arch., p. 1159 (2021)

    Google Scholar 

  58. Schnorr, C.P.: Efficient signature generation by smart cards. J. Cryptol. 4(3), 161–174 (1991). https://doi.org/10.1007/BF00196725

    Article  MathSciNet  MATH  Google Scholar 

  59. 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

    Chapter  Google Scholar 

  60. Tomescu, A., et al.: Towards scalable threshold cryptosystems. In: IEEE S &P, pp. 877–893 (2020)

    Google Scholar 

  61. Vlasov, A., Panarin, K.: Transparent polynomial commitment scheme with polylogarithmic communication complexity. IACR Cryptol. ePrint Arch. 2019, 1020 (2019)

    Google Scholar 

  62. Wahby, R.S., Tzialla, I., Shelat, A., Thaler, J., Walfish, M.: Doubly-efficient zkSNARKs without trusted setup. In: IEEE S &P, pp. 926–943 (2018)

    Google Scholar 

  63. Wails, R., Johnson, A., Starin, D., Yerukhimovich, A., Gordon, S.D.: Stormy: statistics in tor by measuring securely. In: CCS, pp. 615–632 (2019)

    Google Scholar 

  64. Wieder, U.: Balanced allocations with heterogenous bins. In: SPAA, pp. 188–193 (2007)

    Google Scholar 

  65. 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

    Chapter  Google Scholar 

  66. Yuan, J., Yu, S.: Proofs of retrievability with public verifiability and constant communication cost in cloud. In: SCC@ASIACCS, pp. 19–26. ACM (2013)

    Google Scholar 

  67. Zhang, J., Xie, T., Zhang, Y., Song, D.: Transparent polynomial delegation and its applications to zero knowledge proof. In: IEEE S &P (2020)

    Google Scholar 

  68. Zhang, Y., Genkin, D., Katz, J., Papadopoulos, D., Papamanthou, C.: VSQL: verifying arbitrary SQL queries over dynamic outsourced databases. In: IEEE S &P, pp. 863–880 (2017)

    Google Scholar 

  69. Zhang, Y., Genkin, D., Katz, J., Papadopoulos, D., Papamanthou, C.: A zero-knowledge version of VSQL. IACR Cryptol. ePrint Arch. 2017, 1146 (2017)

    Google Scholar 

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Acknowledgements

We thank the anonymous PKC‘23 reviewers for their helpful comments. The first and second authors are supported by ISF grant No. 1316/18. The second, third and fifth authors are supported by DARPA under Contract No. HR001120C0087. The third author was supported by Technology and Humanity Fund from Georgetown University’s McCourt School of Public Policy. 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.

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Authors and Affiliations

  1. Bar Ilan University, Ramat Gan, Israel

    Rishabh Bhadauria & Carmit Hazay

  2. Georgetown University, Washington, DC, USA

    Muthuramakrishnan Venkitasubramaniam

  3. Ligero Inc., Rochester, USA

    Carmit Hazay & Muthuramakrishnan Venkitasubramaniam

  4. Texas A&M University, College Station, USA

    Wenxuan Wu & Yupeng Zhang

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  1. Rishabh Bhadauria
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  2. Carmit Hazay
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  3. Muthuramakrishnan Venkitasubramaniam
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  4. Wenxuan Wu
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Correspondence to Rishabh Bhadauria .

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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

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Bhadauria, R., Hazay, C., Venkitasubramaniam, M., Wu, W., Zhang, Y. (2023). Private Polynomial Commitments and Applications to MPC. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13941. Springer, Cham. https://doi.org/10.1007/978-3-031-31371-4_5

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