Succinct Arguments from Multi-prover Interactive Proofs and Their Efficiency Benefits

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7417)

Abstract

Succinct arguments of knowledge are computationally-sound proofs of knowledge for NP where the verifier’s running time is independent of the time complexity of the NP nondeterministic machine for the considered language.

Existing succinct argument constructions are, typically, based on techniques that combine cryptographic hashing and probabilistically-checkable proofs (PCPs), and thus, in light of today’s state-of-the-art PCP technology, are quite inefficient: either one uses long PCP proofs with lots of redundancy to make the verifier fast but at the cost of making the prover slow, or one uses short PCP proofs to make the prover fast but at the cost of making the verifier slow.

To obtain better efficiency, we propose to investigate the alternative approach of constructing succinct arguments based on multi-prover interactive proofs (MIPs) and stronger cryptographic techniques:

(1) We construct a one-round succinct MIP of knowledge protocol where (i) each prover is highly efficient in terms of time AND space, and ALSO (ii) the verifier is highly efficient.

(2) We show how to transform any one round MIP protocol to a succinct four-message argument (with a single prover), while preserving the time and space efficiency of the original MIP protocol.

As a main tool for this transformation, we construct a succinct multi-function commitment that (a) allows the sender to commit to a vector of functions in time and space complexity that are essentially the same as those needed for a single evaluation of the functions, and (b) ensures that the receiver’s running time is essentially independent of the function. The scheme is based on fully-homomorphic encryption (and no additional assumptions are needed for our succinct argument).

(3) In addition, we revisit the problem of non-interactive succinct arguments of knowledge (SNARKs), where known impossibilities rule out solutions based on black-box reductions to standard assumptions. We formulate a natural (though non-standard) variant of homomorphic encryption that has a homomorphism-extraction property. We then show that his primitive essentially allows to “squash” our interactive protocol, while again preserving time and space efficiency. We further show that this variant is, in fact, implied by the existence of SNARKs.

References

  1. [ABOR00]
    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: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 463–474. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. [BC12]
    Bitansky, N., Chiesa, A.: Succinct arguments from multi-prover interactive proofs and their efficiency benefits. Cryptology ePrint Archive (2012)Google Scholar
  3. [BCC88]
    Brassard, G., Chaum, D., Crépeau, C.: Minimum disclosure proofs of knowledge. Journal of Computer and System Sciences 37(2), 156–189 (1988)Google Scholar
  4. [BCCT11]
    Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again. Cryptology ePrint Archive, Report 2011/443 (2011)Google Scholar
  5. [BCCT12]
    Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: Recursive composition and bootstrapping for snarks and proof-carrying data. Cryptology ePrint Archive, Report 2012/095 (2012)Google Scholar
  6. [BFL90]
    Babai, L., Fortnow, L., Lund, C.: Nondeterministic exponential time has two-prover interactive protocols. In: Proceedings of the 31st Annual Symposium on Foundations of Computer Science, SFCS 1990, pp. 16–25 (1990)Google Scholar
  7. [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, STOC 1991, pp. 21–32 (1991)Google Scholar
  8. [BG08]
    Barak, B., Goldreich, O.: Universal arguments and their applications. SIAMJournal on Computing 38(5), 1661–1694 (2008); Preliminary version appeared in CCC 2002Google Scholar
  9. [BHZ87]
    Boppana, R.B., Håstad, J., Zachos, S.: Does co-NP have short interactive proofs? Information Processing Letters 25(2), 127–132 (1987)Google Scholar
  10. [BOGKW88]
    Ben-Or, M., Goldwasser, S., Kilian, J., Wigderson, A.: Multi-prover interactive proofs: how to remove intractability assumptions. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, STOC 1988, pp. 113–131 (1988)Google Scholar
  11. [BP04a]
    Bellare, M., Palacio, A.: The Knowledge-of-Exponent Assumptions and 3-Round Zero-Knowledge Protocols. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 273–289. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. [BP04b]
    Bellare, M., Palacio, A.: Towards Plaintext-Aware Public-Key Encryption Without Random Oracles. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 48–62. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. [BSCGT12a]
    Ben-Sasson, E., Chiesa, A., Genkin, D., Tromer, E.: Fast reductions from RAMs to delegatable succinct constraint satisfaction problems. Cryptology ePrint Archive, Report Report 2012/071 (2012)Google Scholar
  14. [BSCGT12b]
    Ben-Sasson, E., Chiesa, A., Genkin, D., Tromer, E.: On the concrete-efficiency threshold of probabilistically-checkable proofs. Electronic Colloquium on Computational Complexity, TR12-045 (2012)Google Scholar
  15. [BSGH+05]
    Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Short PCPs verifiable in polylogarithmic time. In: Proceedings of the 20th Annual IEEE Conference on Computational Complexity, CCC 2005, pp. 120–134 (2005)Google Scholar
  16. [BSS08]
    Ben-Sasson, E., Sudan, M.: Short PCPs with polylog query complexity. SIAM Journal on Computing 38(2), 551–607 (2008)Google Scholar
  17. [BSW11]
    Boneh, D., Segev, G., Waters, B.: Targeted malleability: Homomorphic encryption for restricted computations. Cryptology ePrint Archive, Report 2011/311 (2011)Google Scholar
  18. [BV11]
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, FOCS 2011 (2011)Google Scholar
  19. [CHS05]
    Canetti, R., Halevi, S., Steiner, M.: Hardness Amplification of Weakly Verifiable Puzzles. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 17–33. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. [CKV10]
    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)CrossRefGoogle Scholar
  21. [CL10]
    Chung, K.-M., Liu, F.-H.: Parallel Repetition Theorems for Interactive Arguments. In:Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 19–36. Springer, Heidelberg (2010)Google Scholar
  22. [CT10]
    Chiesa, A., Tromer, E.: Proof-carrying data and hearsay arguments from signature cards. In: Proceedings of the 1st Symposium on Innovations in Computer Science, ICS 2010, pp. 310–331 (2010)Google Scholar
  23. [Dam92]
    Damgård, I.: Towards Practical Public Key Systems Secure against Chosen Ciphertext Attacks. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 445–456. Springer, Heidelberg (1992)Google Scholar
  24. [DCL08]
    Di Crescenzo, G., Lipmaa, H.: Succinct NP Proofs from an Extractability Assumption. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 175–185. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  25. [DFH11]
    Damgård, I., Faust, S., Hazay, C.: Secure two-party computation with low communication. Cryptology ePrint Archive, Report 2011/508 (2011)Google Scholar
  26. [DLN+04]
    Dwork, C., Langberg,M., Naor,M., Nissim, K., Reingold, O.: Succinct NP proofs and spooky interactions (December 2004), http://www.openu.ac.il/home/mikel/papers/spooky.ps
  27. [FS87]
    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)CrossRefGoogle Scholar
  28. [GGPR12]
    Gennaro, R., Gentry, C., Parno, B., Raykova, M.: Quadratic span programs and succinct NIZKs without PCPs. Cryptology ePrint Archive, Report 2012/215 (2012)Google Scholar
  29. [GH98]
    Goldreich, O., Håstad, J.: On the complexity of interactive proofs with bounded communication. Information Processing Letters 67(4), 205–214 (1998)Google Scholar
  30. [GLR11]
    Goldwasser, S., Lin, H., Rubinstein, A.: Delegation of computation without rejection problem from designated verifier CS-proofs. Cryptology ePrint Archive, Report 2011/456 (2011)Google Scholar
  31. [GMR89]
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM Journal on Computing 18(1), 186–208 (1989); Preliminary version appeared in STOC 1985Google Scholar
  32. [Gro10]
    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)CrossRefGoogle Scholar
  33. [GVW02]
    Goldreich, O., Vadhan, S., Wigderson, A.: On interactive proofs with a laconic prover. Computational Complexity 11(1/2), 1–53 (2002)Google Scholar
  34. [GW11]
    Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, STOC 2011, pp. 99–108 (2011)Google Scholar
  35. [Hai09]
    Haitner, I.: A parallel repetition theorem for any interactive argument. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pp. 241–250 (2009)Google Scholar
  36. [Har04]
    Harsha, P.: Robust PCPs of Proximity and Shorter PCPs. PhD thesis, MIT, EECS (September 2004)Google Scholar
  37. [IKO07]
    Ishai, Y., Kushilevitz, E., Ostrovsky, R.: Efficient arguments without short PCPs. In: Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007, pp. 278–291 (2007)Google Scholar
  38. [Kil92]
    Kilian, J.: A note on efficient zero-knowledge proofs and arguments. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, STOC 1992, pp. 723–732 (1992)Google Scholar
  39. [Lip11]
    Lipmaa, H.: Progression-free sets and sublinear pairing-based non-interactive zero-knowledge arguments. Cryptology ePrint Archive, Report 2011/009 (2011)Google Scholar
  40. [Mic00]
    Micali, S.: Computationally sound proofs. SIAM Journal on Computing 30(4), 1253–1298 (2000); Preliminary version appeared in FOCS 1994Google Scholar
  41. [MR08]
    Moshkovitz, D., Raz, R.: Two-query PCP with subconstant error. Journal of the ACM 57, 1–29 (2008); Preliminary version appeared in FOCS 2008Google Scholar
  42. [Nao03]
    Naor, M.: On Cryptographic Assumptions and Challenges. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 96–109. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  43. [RS97]
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, STOC 1997, pp. 475–484 (1997)Google Scholar
  44. [RV09]
    Rothblum, G.N., Vadhan, S.: Are PCPs inherent in efficient arguments? In: Proceedings of the 24th IEEE Annual Conference on Computational Complexity, CCC 2009, pp. 81–92 (2009)Google Scholar
  45. [Sha92]
    Shamir, A.: IP = PSPACE. Journal of the ACM 39(4), 869–877 (1992)Google Scholar
  46. [TS96]
    Ta-Shma, A.: A note on PCP vs.MIP. Information Processing Letters 58, 135–140 (1996)Google Scholar
  47. [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)CrossRefGoogle Scholar
  48. [Wee05]
    Wee, H.: On Round-Efficient Argument Systems. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 140–152. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2012 2012

Authors and Affiliations

  1. 1.TAUTel AvivIsrael
  2. 2.MITCambridgeUSA

Personalised recommendations