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Incrementally Verifiable Computation via Incremental PCPs

  • Moni NaorEmail author
  • Omer Paneth
  • Guy N. Rothblum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11892)

Abstract

If I commission a long computation, how can I check that the result is correct without re-doing the computation myself? This is the question that efficient verifiable computation deals with. In this work, we address the issue of verifying the computation as it unfolds. That is, at any intermediate point in the computation, I would like to see a proof that the current state is correct. Ideally, these proofs should be short, non-interactive, and easy to verify. In addition, the proof at each step should be generated efficiently by updating the previous proof, without recomputing the entire proof from scratch. This notion, known as incrementally verifiable computation, was introduced by Valiant [TCC 08] about a decade ago. Existing solutions follow the approach of recursive proof composition and can be based on strong and non-falsifiable cryptographic assumptions (so-called “knowledge assumptions”).

In this work, we present a new framework for constructing incrementally verifiable computation schemes in both the publicly verifiable and designated-verifier settings. Our designated-verifier scheme is based on somewhat homomorphic encryption (which can be based on Learning with Errors) and our publicly verifiable scheme is based on the notion of zero-testable homomorphic encryption, which can be constructed from ideal multi-linear maps [Paneth and Rothblum, TCC 17].

Our framework is anchored around the new notion of a probabilistically checkable proof (PCP) with incremental local updates. An incrementally updatable PCP proves the correctness of an ongoing computation, where after each computation step, the value of every symbol can be updated locally without reading any other symbol. This update results in a new PCP for the correctness of the next step in the computation. Our primary technical contribution is constructing such an incrementally updatable PCP. We show how to combine updatable PCPs with recently suggested (ordinary) verifiable computation to obtain our results.

References

  1. [BCC+17]
    Bitansky, N., et al.: The hunting of the SNARK. J. Cryptol. 30(4), 989–1066 (2017)MathSciNetCrossRefGoogle Scholar
  2. [BCCT13]
    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
  3. [BFLS91]
    Babai, L., Fortnow, L., Levin, L.A., Szegedy, M.: Checking computations in polylogarithmic time. In: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, New Orleans, Louisiana, USA, 5–8 May 1991, pp. 21–31 (1991)Google Scholar
  4. [BHK16]
    Brakerski, Z., Holmgren, J., Kalai, Y.T.: Non-interactive RAM and batch NP delegation from any PIR. IACR Cryptology ePrint Archive, 2016:459 (2016)Google Scholar
  5. [BHK17]
    Brakerski, Z., Holmgren, J., Kalai, Y.T.: Non-interactive delegation and batch NP verification from standard computational assumptions. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, 19–23 June 2017, pp. 474–482 (2017)Google Scholar
  6. [BMW98]
    Biehl, I., Meyer, B., Wetzel, S.: Ensuring the integrity of agent-based computations by short proofs. In: Rothermel, K., Hohl, F. (eds.) MA 1998. LNCS, vol. 1477, pp. 183–194. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0057658CrossRefGoogle Scholar
  7. [BV11]
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22–25, 2011, pp. 97–106 (2011)Google Scholar
  8. [DHRW16]
    Dodis, Y., Halevi, S., Rothblum, R.D., Wichs, D.: Spooky encryption and its applications. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 93–122. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53015-3_4CrossRefGoogle Scholar
  9. [DLN+00]
    Dwork, C., Langberg, M., Naor, M., Nissim, K., Reingold, O.: Succinct proofs for \(\sf NP\) ander spooky interactions. Manuscript (2000). http://www.wisdom.weizmann.ac.il/~naor/PAPERS/spooky.pdf
  10. [DNR16]
    Dwork, C., Naor, M., Rothblum, G.N.: Spooky interaction and its discontents: compilers for succinct two-message argument systems. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 123–145. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53015-3_5CrossRefGoogle Scholar
  11. [Gen09]
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, 31 May–2 June, pp. 169–178 (2009)Google Scholar
  12. [GH98]
    Goldreich, O., Håstad, J.: On the complexity of interactive proofs with bounded communication. Inf. Process. Lett. 67(4), 205–214 (1998)MathSciNetCrossRefGoogle Scholar
  13. [GSW13]
    Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40041-4_5CrossRefGoogle Scholar
  14. [HR18]
    Holmgren, J., Rothblum, R.: Delegating computations with (almost) minimal time and space overhead. In: 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, 7–9 October 2018, pp. 124–135 (2018)Google Scholar
  15. [KPY19]
    Kalai, Y., Paneth, O., Yang, L.: How to delegate computations publicly. In: STOC (2019)Google Scholar
  16. [KRR13]
    Kalai, Y.T., Raz, R., Rothblum, R.D.: Delegation for bounded space. In: STOC, pp. 565–574 (2013)Google Scholar
  17. [KRR14]
    Kalai, Y.T., Raz, R., Rothblum, R.D.: How to delegate computations: the power of no-signaling proofs. In: Symposium on Theory of Computing, STOC 2014, New York, NY, USA, 31 May–03 June 2014, pp. 485–494 (2014)Google Scholar
  18. [PR17]
    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_9CrossRefGoogle Scholar
  19. [Val08]
    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_1CrossRefzbMATHGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.MIT and Northeastern UniversityCambridgeUSA

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