Advertisement

No-signaling Linear PCPs

  • Susumu Kiyoshima
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11239)

Abstract

In this paper, we give a no-signaling linear probabilistically checkable proof (PCP) system for polynomial-time deterministic computation, i.e., a PCP system for \({\mathcal {P}}\) such that (1) the PCP oracle is a linear function and (2) the soundness holds against any (computational) no-signaling cheating prover, who is allowed to answer each query according to a distribution that depends on the entire query set in a certain way. To the best of our knowledge, our construction is the first PCP system that satisfies these two properties simultaneously.

As an application of our PCP system, we obtain a 2-message scheme for delegating computation by using a known transformation. Compared with existing 2-message delegation schemes based on standard cryptographic assumptions, our scheme requires preprocessing but has a simpler structure and makes use of different (possibly cheaper) standard cryptographic primitives, namely additive/multiplicative homomorphic encryption schemes.

References

  1. 1.
    Aiello, W., Bhatt, S., Ostrovsky, R., Rajagopalan, S.R.: Fast verification of any remote procedure call: short witness-indistinguishable one-round proofs for NP. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 463–474. Springer, Heidelberg (2000).  https://doi.org/10.1007/3-540-45022-X_39 CrossRefGoogle Scholar
  2. 2.
    Ananth, P., Chen, Y.-C., Chung, K.-M., Lin, H., Lin, W.-K.: Delegating RAM computations with adaptive soundness and privacy. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 3–30. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53644-5_1 CrossRefGoogle Scholar
  3. 3.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009). http://theory.cs.princeton.edu/complexity/book.pdf
  4. 4.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM 45(1), 70–122 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Babai, L., Fortnow, L., Levin, L.A., Szegedy, M.: Checking computations in polylogarithmic time. In: 23rd ACM STOC, pp. 21–31. ACM Press, May 1991Google Scholar
  7. 7.
    Badrinarayanan, S., Kalai, Y.T., Khurana, D., Sahai, A., Wichs, D.: Succinct delegation for low-space non-deterministic computation. In: Diakonikolas, I., Kempe, D., Henzinger, M. (eds.) 50th ACM STOC, pp. 709–721. ACM Press, June 2018Google Scholar
  8. 8.
    Barak, B., Goldreich, O.: Universal arguments and their applications. SIAM J. Comput. 38(5), 1661–1694 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ben-Sasson, E., Chiesa, A., Genkin, D., Tromer, E., Virza, M.: SNARKs for C: verifying program executions succinctly and in zero knowledge. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 90–108. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40084-1_6 CrossRefzbMATHGoogle Scholar
  10. 10.
    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 CrossRefGoogle Scholar
  11. 11.
    Ben-Sasson, E., Chiesa, A., Tromer, E., Virza, M.: Succinct non-interactive zero knowledge for a von Neumann architecture. In: Proceedings of the 23rd USENIX Security Symposium, San Diego, CA, USA, 20–22 August 2014, pp. 781–796 (2014)Google Scholar
  12. 12.
    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/BFb0057658 CrossRefGoogle Scholar
  13. 13.
    Bitansky, N., et al.: The hunting of the SNARK. J. Cryptol. 30(4), 989–1066 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: Recursive composition and bootstrapping for SNARKS and proof-carrying data. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th ACM STOC, pp. 111–120. ACM Press, June 2013Google Scholar
  15. 15.
    Bitansky, N., Chiesa, A.: Succinct arguments from multi-prover interactive proofs and their efficiency benefits. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 255–272. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_16 CrossRefGoogle Scholar
  16. 16.
    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 CrossRefGoogle Scholar
  17. 17.
    Bitansky, N., Garg, S., Lin, H., Pass, R., Telang, S.: Succinct randomized encodings and their applications. In: Servedio, R.A., Rubinfeld, R. (eds.) 47th ACM STOC, pp. 439–448. ACM Press, June 2015Google Scholar
  18. 18.
    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 CrossRefGoogle Scholar
  19. 19.
    Boneh, D., Ishai, Y., Sahai, A., Wu, D.J.: Lattice-based SNARGs and their application to more efficient obfuscation. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 247–277. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56617-7_9 CrossRefGoogle Scholar
  20. 20.
    Brakerski, Z., Holmgren, J., Kalai, Y.T.: Non-interactive delegation and batch NP verification from standard computational assumptions. In: Hatami, H., McKenzie, P., King, V. (eds.) 49th ACM STOC, pp. 474–482. ACM Press, June 2017Google Scholar
  21. 21.
    Canetti, R., Chen, Y., Holmgren, J., Raykova, M.: Adaptive succinct garbled RAM or: how to delegate your database. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 61–90. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53644-5_3 CrossRefGoogle Scholar
  22. 22.
    Canetti, R., Holmgren, J.: Fully succinct garbled RAM. In: Sudan, M. (ed.) ITCS 2016, pp. 169–178. ACM, January 2016Google Scholar
  23. 23.
    Canetti, R., Holmgren, J., Jain, A., Vaikuntanathan, V.: Succinct garbling and indistinguishability obfuscation for RAM programs. In: Servedio, R.A., Rubinfeld, R. (eds.) 47th ACM STOC, pp. 429–437. ACM Press, June 2015Google Scholar
  24. 24.
    Chen, Y.C., Chow, S.S.M., Chung, K.M., Lai, R.W.F., Lin, W.K., Zhou, H.S.: Cryptography for parallel RAM from indistinguishability obfuscation. In: Sudan, M. (ed.) ITCS 2016, pp. 179–190. ACM, January 2016Google Scholar
  25. 25.
    Chiesa, A., Manohar, P., Shinkar, I.: Probabilistic checking against non-signaling strategies from linearity testing. Electronic Colloquium on Computational Complexity (ECCC), TR18-123 (2018). https://eccc.weizmann.ac.il/report/2018/123/
  26. 26.
    Chiesa, A., Manohar, P., Shinkar, I.: Testing linearity against non-signaling strategies. In: 33rd Computational Complexity Conference, CCC 2018, San Diego, California, USA, 22–24 June 2018, pp. 17:1–17:37 (2018)Google Scholar
  27. 27.
    Chung, K.-M., Kalai, Y., Vadhan, S.: Improved delegation of computation using fully homomorphic encryption. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 483–501. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-14623-7_26 CrossRefGoogle Scholar
  28. 28.
    Damgård, I., Faust, S., Hazay, C.: Secure two-party computation with low communication. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 54–74. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-28914-9_4 CrossRefGoogle Scholar
  29. 29.
    Danezis, G., Fournet, C., Groth, J., Kohlweiss, M.: Square span programs with applications to succinct NIZK arguments. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8873, pp. 532–550. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-45611-8_28 CrossRefGoogle Scholar
  30. 30.
    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 CrossRefGoogle Scholar
  31. 31.
    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 CrossRefGoogle Scholar
  32. 32.
    Goldwasser, S., Kalai, Y.T., Rothblum, G.N.: Delegating computation: interactive proofs for muggles. In: Ladner, R.E., Dwork, C. (eds.) 40th ACM STOC, pp. 113–122. ACM Press, May 2008Google Scholar
  33. 33.
    Goldwasser, S., Micali, S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Groth, J.: Short pairing-based non-interactive zero-knowledge arguments. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 321–340. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-17373-8_19 CrossRefGoogle Scholar
  35. 35.
    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 CrossRefGoogle Scholar
  36. 36.
    Holmgren, J., Rothblum, R.: Delegating computations with (almost) minimal time and space overhead. Electronic Colloquium on Computational Complexity (ECCC), TR18-161 (2018). https://eccc.weizmann.ac.il/report/2018/161/. To appear at FOCS 2018
  37. 37.
    Ishai, Y., Kushilevitz, E., Ostrovsky, R.: Efficient arguments without short PCPs. In: 22nd Annual IEEE Conference on Computational Complexity, CCC 2007, San Diego, California, USA, 13–16 June 2007, pp. 278–291 (2007)Google Scholar
  38. 38.
    Kalai, Y., Paneth, O.: Delegating RAM computations. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 91–118. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53644-5_4 CrossRefGoogle Scholar
  39. 39.
    Kalai, Y.T., Raz, R., Rothblum, R.D.: Delegation for bounded space. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th ACM STOC, pp. 565–574. ACM Press, June 2013Google Scholar
  40. 40.
    Kalai, Y.T., Raz, R., Rothblum, R.D.: How to delegate computations: the power of no-signaling proofs. In: Shmoys, D.B. (ed.) 46th ACM STOC, pp. 485–494. ACM Press, May/Jun 2014Google Scholar
  41. 41.
    Kilian, J.: A note on efficient zero-knowledge proofs and arguments (extended abstract). In: 24th ACM STOC, pp. 723–732. ACM Press, May 1992Google Scholar
  42. 42.
    Kiyoshima, S.: No-signaling linear PCPs. Cryptology ePrint Archive, Report 2018/649 (2018). https://eprint.iacr.org/2018/649
  43. 43.
    Koppula, V., Lewko, A.B., Waters, B.: Indistinguishability obfuscation for Turing machines with unbounded memory. In: Servedio, R.A., Rubinfeld, R. (eds.) 47th ACM STOC, pp. 419–428. ACM Press, June 2015Google Scholar
  44. 44.
    Lipmaa, H.: Progression-free sets and sublinear pairing-based non-interactive zero-knowledge arguments. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 169–189. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-28914-9_10 CrossRefGoogle Scholar
  45. 45.
    Lipmaa, H.: Succinct non-interactive zero knowledge arguments from span programs and linear error-correcting codes. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013. LNCS, vol. 8269, pp. 41–60. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-42033-7_3 CrossRefGoogle Scholar
  46. 46.
    Micali, S.: Computationally sound proofs. SIAM J. Comput. 30(4), 1253–1298 (2000)MathSciNetCrossRefGoogle Scholar
  47. 47.
    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_9 CrossRefGoogle Scholar
  48. 48.
    Parno, B., Raykova, M., Vaikuntanathan, V.: How to delegate and verify in public: verifiable computation from attribute-based encryption. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 422–439. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-28914-9_24 CrossRefGoogle Scholar
  49. 49.
    Setty, S., Blumberg, A.J., Walfish, M.: Toward practical and unconditional verification of remote computations. In: Workshop on Hot Topics in Operating Systems (HotOS). USENIX - Advanced Computing Systems Association (2011)Google Scholar
  50. 50.
    Setty, S., Braun, B., Vu, V., Blumberg, A.J., Parno, B., Walfish, M.: Resolving the conflict between generality and plausibility in verified computation. In: Proceedings of the ACM European Conference on Computer Systems (EuroSys). ACM, April 2013Google Scholar
  51. 51.
    Setty, S.T.V., McPherson, R., Blumberg, A.J., Walfish, M.: Making argument systems for outsourced computation practical (sometimes). In: NDSS 2012. The Internet Society, February 2012Google Scholar
  52. 52.
    Setty, S.T.V., Vu, V., Panpalia, N., Braun, B., Blumberg, A.J., Walfish, M.: Taking proof-based verified computation a few steps closer to practicality. In: Proceedings of the 21th USENIX Security Symposium, Bellevue, WA, USA, 8–10 August 2012, pp. 253–268 (2012)Google Scholar
  53. 53.
    Vu, V., Setty, S.T.V., Blumberg, A.J., Walfish, M.: A hybrid architecture for interactive verifiable computation. In: 2013 IEEE Symposium on Security and Privacy, pp. 223–237. IEEE Computer Society Press, May 2013Google Scholar

Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  1. 1.NTT Secure Platform LaboratoriesTokyoJapan

Personalised recommendations