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Algebraic Reductions of Knowledge

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14084))

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Abstract

We introduce reductions of knowledge, a generalization of arguments of knowledge, which reduce checking knowledge of a witness in one relation to checking knowledge of a witness in another (simpler) relation. Reductions of knowledge unify a growing class of modern techniques as well as provide a compositional framework to modularly reason about individual steps in complex arguments of knowledge. As a demonstration, we simplify and unify recursive arguments over linear algebraic statements by decomposing them as a sequence of reductions of knowledge. To do so, we develop the tensor reduction of knowledge, which generalizes the central reductive step common to many recursive arguments. Underlying the tensor reduction of knowledge is a new information-theoretic reduction, which, for any modules U, \(U_1\), and \(U_2\) such that \(U \cong U_1 \otimes U_2\), reduces the task of evaluating a homomorphism in U to evaluating a homomorphism in \(U_1\) and evaluating a homomorphism in \(U_2\).

Abhiram Kothapalli was supported by a fellowship from Protocol Labs, a gift from Bosch, NSF Grant No. 1801369, and the CONIX Research Center, one of six centers in JUMP, a Semiconductor Research Corporation program sponsored by DARPA. An extended version of this work is available on the Cryptology ePrint Archive [30].

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Notes

  1. 1.

    We recommend Bitansky et al. [8, Remark 6.3] for details on such assumptions.

  2. 2.

    The tensor relation can be formally understood as a ternary relation where any public parameters are ignored. This makes it compatible with the reductions of knowledge framework which works over ternary relations defined over public parameter, statement, and witness tuples.

References

  1. Abe, M., Fuchsbauer, G., Groth, J., Haralambiev, K., Ohkubo, M.: Structure-preserving signatures and commitments to group elements. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 209–236. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_12

    Chapter  Google Scholar 

  2. Arora, S., Barak, B.: Computational complexity: a modern approach. Cambridge University Press (2009)

    Google Scholar 

  3. Attema, T., Cramer, R.: Compressed \(\Sigma \)-protocol theory and practical application to plug  & play secure algorithmics. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12172, pp. 513–543. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56877-1_18

  4. Attema, T., Cramer, R., Rambaud, M.: Compressed \(\Sigma \)-protocols for bilinear group arithmetic circuits and application to logarithmic transparent threshold signatures. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021. LNCS, vol. 13093, pp. 526–556. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92068-5_18

  5. Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: Proceedings of the 1st ACM Conference on Computer and Communications Security, pp. 62–73 (1993)

    Google Scholar 

  6. 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

    Chapter  Google Scholar 

  7. Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, pp. 326–349 (2012)

    Google Scholar 

  8. Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: Recursive composition and bootstrapping for SNARKs and proof-carrying data. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp. 111–120 (2013)

    Google Scholar 

  9. Boneh, D., Drake, J., Fisch, B., Gabizon, A.: Halo Infinite: proof-carrying data from additive polynomial commitments. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12825, pp. 649–680. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84242-0_23

    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. LNCS, vol. 9666, pp. 327–357. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_12

    Chapter  MATH  Google Scholar 

  11. Bootle, J., Chiesa, A., Sotiraki, K.: Sumcheck arguments and their applications. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12825, pp. 742–773. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84242-0_26

    Chapter  Google Scholar 

  12. Bowe, S., Grigg, J., Hopwood, D.: Recursive proof composition without a trusted setup. Cryptology ePrint Archive, Paper 2019/1021 (2019)

    Google Scholar 

  13. 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 (SP), pp. 315–334. IEEE (2018)

    Google Scholar 

  14. Bünz, B., Chiesa, A., Lin, W., Mishra, P., Spooner, N.: Proof-carrying data without succinct arguments. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12825, pp. 681–710. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84242-0_24

    Chapter  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. 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

    Chapter  Google Scholar 

  18. Campanelli, M., Nitulescu, A., Ràfols, C., Zacharakis, A., Zapico, A.: Linear-map vector commitments and their practical applications. In: Agrawal, S., Lin, D. (eds.) Advances in Cryptology - ASIACRYPT 2022, pp. 189–219. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-22972-5_7

    Chapter  Google Scholar 

  19. 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 

  20. Chung, H., Han, K., Ju, C., Kim, M., Seo, J.H.: Bulletproofs+: shorter proofs for a privacy-enhanced distributed ledger. IEEE Access 10, 42067–42082 (2022)

    Article  Google Scholar 

  21. Delignat-Lavaud, A., Fournet, C., Kohlweiss, M., Parno, B.: Cinderella: turning shabby X. 509 certificates into elegant anonymous credentials with the magic of verifiable computation. In: 2016 IEEE Symposium on Security and Privacy (SP), pp. 235–254. IEEE (2016)

    Google Scholar 

  22. 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 

  23. Fuchsbauer, G., Kiltz, E., Loss, J.: The algebraic group model and its applications. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 33–62. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_2

    Chapter  Google Scholar 

  24. 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)

    Google Scholar 

  25. 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 

  26. Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: Proceedings of the forty-third annual ACM symposium on Theory of computing, pp. 99–108 (2011)

    Google Scholar 

  27. Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof-systems. In: Providing Sound Foundations for Cryptography: On the Work of Shafi Goldwasser and Silvio Micali, pp. 203–225 (2019)

    Google Scholar 

  28. Kosba, A., Miller, A., Shi, E., Wen, Z., Papamanthou, C.: Hawk: the blockchain model of cryptography and privacy-preserving smart contracts. In: 2016 IEEE Symposium on Security and Privacy (SP), pp. 839–858. IEEE (2016)

    Google Scholar 

  29. Kothapalli, A., Masserova, E., Parno, B.: Poppins: A direct construction for asymptotically optimal zkSNARKs. Cryptology ePrint Archive, Report 2020/1318 (2020)

    Google Scholar 

  30. Kothapalli, A., Parno, B.: Algebraic reductions of knowledge. Cryptology ePrint Archive, Paper 2022/009 (2022)

    Google Scholar 

  31. Kothapalli, A., Setty, S., Tzialla, I.: Nova: Recursive zero-knowledge arguments from folding schemes. In: Dodis, Y., Shrimpton, T. (eds.) CRYPTO 2022, Part IV, pp. 359–388. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15985-5_13

  32. 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

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. 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 

  35. Ràfols, C., Zapico, A.: An algebraic framework for universal and updatable SNARKs. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12825, pp. 774–804. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84242-0_27

    Chapter  Google Scholar 

  36. Ràfols, C., Zacharakis, A.: Folding schemes with selective verification. Cryptology ePrint Archive, Paper 2022/1576 (2022)

    Google Scholar 

  37. Sasson, E.B., et al.: Zerocash: decentralized anonymous payments from bitcoin. In: 2014 IEEE Symposium on Security and Privacy, pp. 459–474. IEEE (2014)

    Google Scholar 

  38. Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM (JACM) 27(4), 701–717 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  39. 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 

  40. Tzialla, I., Kothapalli, A., Parno, B., Setty, S.: Transparency dictionaries with succinct proofs of correct operation. In: Network and Distributed System Security (NDSS) 2022, April 2022

    Google Scholar 

  41. Valiant, P.: Incrementally verifiable computation or proofs of knowledge imply time/space efficiency. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 1–18. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78524-8_1

    Chapter  MATH  Google Scholar 

  42. 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 (SP), pp. 926–943. IEEE (2018)

    Google Scholar 

  43. Zhang, Y., Katz, J., Papamanthou, C.: IntegriDB: verifiable SQL for outsourced databases. In: Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security, pp. 1480–1491 (2015)

    Google Scholar 

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Acknowledgments

We thank Jonathan Bootle, Quang Dao, Vipul Goyal, Yael Tauman Kalai, Jonathan Lee, Srinath Setty, Elaine Shi, and Zoe Wellner for comments on earlier versions of this work.

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  1. Carnegie Mellon University, Pittsburgh, USA

    Abhiram Kothapalli & Bryan Parno

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Correspondence to Abhiram Kothapalli .

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    Helena Handschuh

  2. Brown University, Providence, RI, USA

    Anna Lysyanskaya

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Kothapalli, A., Parno, B. (2023). Algebraic Reductions of Knowledge. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14084. Springer, Cham. https://doi.org/10.1007/978-3-031-38551-3_21

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