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Linear-Map Vector Commitments and Their Practical Applications

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Advances in Cryptology – ASIACRYPT 2022 (ASIACRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13794))

Abstract

Vector commitments (VC) are a cryptographic primitive that allows one to commit to a vector and then “open” some of its positions efficiently. Vector commitments are increasingly recognized as a central tool to scale highly decentralized networks of large size and whose content is dynamic. In this work, we examine the demands on the properties that a vector commitment should satisfy in the light of the emerging plethora of practical applications and propose new constructions that improve the state-of-the-art in several dimensions and offer new tradeoffs. We also propose a unifying framework that captures several constructions and we show how to generically achieve some properties from more basic ones. On the practical side, we focus on building efficient schemes that do not require a new trusted setup (we can reuse existing ceremonies for other pairing-based schemes, such as “powers of tau” run by real-world systems such as Zcash or Filecoin).

Arantxa Zapico has been funded by a Protocol Labs PhD Fellowship PL-RGP1-2021-062. Alexandros Zacharakis has been partially funded by Protocol Labs Research Grant PL-RGP1-2021-048.

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Notes

  1. 1.

    For the applications considered in this work, hiding properties are not necessary. In particular, our commitments are deterministic.

  2. 2.

    We prefer LVC rather than LMC to emphasize the Vector Commitment aspect of our notion.

  3. 3.

    https://developer.mozilla.org/en-US/docs/Web/HTTP/Link_prefetching_FAQ.

  4. 4.

    This remains true even if many setups are updatable [18] and they can be generated and updated non-interactively in a secure way as long as one party is honest. There might be issues if not enough parties participate in generating the SRS or updates are not properly validated.

  5. 5.

    If one uses the Inner Pairing Product argument of Bünz et al. [6] on top of PST commitments as suggested in Hyperproofs the difference in proof size is not so relevant, but IPP will be much cheaper to run.

  6. 6.

    E.g., the one used by ZCash. https://z.cash or and Filecoin [12].

  7. 7.

    The algorithms can be generalized for more proofs. Proof size remains the same, also for cross-commitment aggregation.

  8. 8.

    Naturally, this can be seen as a particular case of unbounded aggregation.

  9. 9.

    This notion can be generalized to more than one position.

  10. 10.

    We use linear forms for simplicity, one could also consider general linear functions.

  11. 11.

    Note that this notation is different than the one we used in the multivariate case. In the latter case, this notation denoted prefixes while here it denotes suffixes. We do this because in each case the corresponding notation makes presentation easier.

  12. 12.

    We assume as in Sect. 5 that at most \(m-1\) powers of \(\tau \) are in the SRS in group \(\mathbb {G}_1\). The degree check is meant to ensure that R(X) is of degree at most \(2^{\kappa }-2\).

References

  1. Aranha, D.F., Pagnin, E., Rodríguez-Henríquez, F.: LOVE a pairing. In: Longa, P., Ràfols, C. (eds.) LATINCRYPT 2021. LNCS, vol. 12912, pp. 320–340. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-88238-9_16

    Chapter  Google Scholar 

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

    Chapter  Google Scholar 

  3. Boneh, D., Boyen, X.: Efficient selective identity-based encryption without random oracles. J. Cryptol. 24(4), 659–693 (2011). https://doi.org/10.1007/s00145-010-9078-6

    Article  MATH  Google Scholar 

  4. Boneh, D., Bünz, B., Fisch, B.: Batching techniques for accumulators with applications to IOPs and stateless blockchains. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019, Part I. LNCS, vol. 11692, pp. 561–586. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_20

    Chapter  Google Scholar 

  5. Bowe, S.: BLS12-381: New zk-SNARK elliptic curve construction. Zcash Company Blog (2017). https://z.cash/blog/new-snark-curve

  6. 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, Part III. LNCS, vol. 13092, pp. 65–97. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92078-4_3

    Chapter  Google Scholar 

  7. Campanelli, M., Fiore, D., Greco, N., Kolonelos, D., Nizzardo, L.: Incrementally Aggregatable Vector Commitments and Applications to Verifiable Decentralized Storage. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020, Part II. LNCS, vol. 12492, pp. 3–35. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_1

    Chapter  Google Scholar 

  8. Campanelli, M., Fiore, D., Han, S., Kim, J., Kolonelos, D., Oh, H.: Succinct zero-knowledge batch proofs for set accumulators. Cryptology ePrint Archive (2021)

    Google Scholar 

  9. 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 

  10. Chepurnoy, A., Papamanthou, C., Zhang, Y.: EDRAX: a cryptocurrency with stateless transaction validation. Cryptology ePrint Archive, Report 2018/968 (2018). https://eprint.iacr.org/2018/968

  11. Daza, V., Ràfols, C., Zacharakis, A.: Updateable inner product argument with logarithmic verifier and applications. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020, Part I. LNCS, vol. 12110, pp. 527–557. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45374-9_18

    Chapter  Google Scholar 

  12. Filecoin: Filecoin powers of tau ceremony attestations (2020). https://github.com/arielgabizon/perpetualpowersoftau

  13. Fisch, B.: PoReps: Proofs of space on useful data. Cryptology ePrint Archive, Report 2018/678 (2018). https://eprint.iacr.org/2018/678

  14. Fuchsbauer, G., Kiltz, E., Loss, J.: The Algebraic Group Model and its Applications. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 33–62. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_2

    Chapter  Google Scholar 

  15. Gorbunov, S., Reyzin, L., Wee, H., Zhang, Z.: Pointproofs: aggregating proofs for multiple vector commitments. In: Ligatti, J., Ou, X., Katz, J., Vigna, G. (eds.) ACM CCS 2020, pp. 2007–2023. ACM Press, November 2020

    Google Scholar 

  16. Grassi, L., Khovratovich, D., Rechberger, C., Roy, A., Schofnegger, M.: Poseidon: a new hash function for \(\{\)Zero-Knowledge\(\}\) proof systems. In: 30th USENIX Security Symposium (USENIX Security 2021), pp. 519–535 (2021)

    Google Scholar 

  17. 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 

  18. Groth, J., Kohlweiss, M., Maller, M., Meiklejohn, S., Miers, I.: Updatable and universal common reference strings with applications to zk-SNARKs. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part III. LNCS, vol. 10993, pp. 698–728. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_24

    Chapter  Google Scholar 

  19. 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 

  20. Lai, R.W.F., Malavolta, G.: Subvector Commitments with Application to Succinct Arguments. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019, Part I. LNCS, vol. 11692, pp. 530–560. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_19

    Chapter  Google Scholar 

  21. Libert, B., Ramanna, S.C., Yung, M.: Functional commitment schemes: from polynomial commitments to pairing-based accumulators from simple assumptions. In: Chatzigiannakis, I., Mitzenmacher, M., Rabani, Y., Sangiorgi, D. (eds.) ICALP 2016, Volume 55 of LIPIcs, pp. 30:1–30:14. Schloss Dagstuhl, July 2016

    Google Scholar 

  22. Libert, B., Yung, M.: Concise mercurial vector commitments and independent zero-knowledge sets with short proofs. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 499–517. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11799-2_30

    Chapter  Google Scholar 

  23. Merkle, R.C.: A digital signature based on a conventional encryption function. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 369–378. Springer, Heidelberg (1988). https://doi.org/10.1007/3-540-48184-2_32

    Chapter  Google Scholar 

  24. 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 

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

    Chapter  Google Scholar 

  26. Srinivasan, S., Chepurnoy, A., Papamanthou, C., Tomescu, A., Zhang, Y.: Hyperproofs: aggregating and maintaining proofs in vector commitments. In: 31st USENIX Security Symposium (USENIX Security 2022), Boston, MA. USENIX Association, August 2022

    Google Scholar 

  27. Tomescu, A., Abraham, I., Buterin, V., Drake, J., Feist, D., Khovratovich, D.: Aggregatable subvector commitments for stateless cryptocurrencies. In: Galdi, C., Kolesnikov, V. (eds.) SCN 2020. LNCS, vol. 12238, pp. 45–64. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-57990-6_3

    Chapter  MATH  Google Scholar 

  28. Tomescu, A., et al.: Towards scalable threshold cryptosystems. In: 2020 IEEE Symposium on Security and Privacy, pp. 877–893. IEEE Computer Society Press, May 2020

    Google Scholar 

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Authors and Affiliations

  1. Protocol Labs, San Francisco, USA

    Matteo Campanelli & Anca Nitulescu

  2. Universitat Pompeu Fabra, Barcelona, Spain

    Carla Ràfols, Alexandros Zacharakis & Arantxa Zapico

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  1. Matteo Campanelli
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Correspondence to Matteo Campanelli .

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Editors and Affiliations

  1. Indian Institute of Technology Madras, Chennai, India

    Shweta Agrawal

  2. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

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Campanelli, M., Nitulescu, A., Ràfols, C., Zacharakis, A., Zapico, A. (2022). Linear-Map Vector Commitments and Their Practical Applications. In: Agrawal, S., Lin, D. (eds) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol 13794. Springer, Cham. https://doi.org/10.1007/978-3-031-22972-5_7

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