Advertisement

Succinct Non-interactive Arguments via Linear Interactive Proofs

  • Nir Bitansky
  • Alessandro Chiesa
  • Yuval Ishai
  • Omer Paneth
  • Rafail Ostrovsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7785)

Abstract

Succinct non-interactive arguments (SNARGs) enable verifying NP statements with lower complexity than required for classical NP verification. Traditionally, the focus has been on minimizing the length of such arguments; nowadays researches have focused also on minimizing verification time, by drawing motivation from the problem of delegating computation.

A common relaxation is a preprocessing SNARG, which allows the verifier to conduct an expensive offline phase that is independent of the statement to be proven later. Recent constructions of preprocessing SNARGs have achieved attractive features: they are publicly-verifiable, proofs consist of only O(1) encrypted (or encoded) field elements, and verification is via arithmetic circuits of size linear in the NP statement. Additionally, these constructions seem to have “escaped the hegemony” of probabilistically-checkable proofs (PCPs) as a basic building block of succinct arguments.

Keywords

Encryption Scheme Homomorphic Encryption Arithmetic Circuit Interactive Proof Argument System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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. [AIK10]
    Applebaum, B., Ishai, Y., Kushilevitz, E.: From Secrecy to Soundness: Efficient Verification via Secure Computation. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 152–163. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. [ALM+98]
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998); Preliminary version in FOCS 1992MathSciNetCrossRefzbMATHGoogle Scholar
  4. [BC12]
    Bitansky, N., Chiesa, A.: Succinct arguments from multi-prover interactive proofs and their efficiency benefits. In: Safavi-Naini, R. (ed.) CRYPTO 2012. LNCS, vol. 7417, pp. 255–272. Springer, Heidelberg (2012)Google Scholar
  5. [BCC88]
    Brassard, G., Chaum, D., Crépeau, C.: Minimum disclosure proofs of knowledge. Journal of Computer and System Sciences 37(2), 156–189 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BCCT12a]
    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 (ITCS 2012), pp. 326–349 (2012)Google Scholar
  7. [BCCT12b]
    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
  8. [BCI+12]
    Bitansky, N., Chiesa, A., Ishai, Y., Ostrovsky, R., Omer, P.: Succinct non-interactive arguments via linear interactive proofs. Cryptology ePrint Archive, Report 2012 (2012)Google Scholar
  9. [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
  10. [BG08]
    Barak, B., GoldreichUniversal, O.: arguments and their applications. SIAM Journal on Computing 38(5), 1661–1694 (2008); Preliminary version appeared in CCC 2002MathSciNetCrossRefzbMATHGoogle Scholar
  11. [BGV11]
    Benabbas, S., Gennaro, R., Vahlis, Y.: Verifiable Delegation of Computation over Large Datasets. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 111–131. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. [BHZ87]
    Boppana, R.B., Håstad, J., Zachos, S.: Does co-NP have short interactive proofs? Information Processing Letters 25(2), 127–132 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [BP04]
    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
  14. [BSCGT12]
    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. [BSHLM09]
    Ben-Sasson, E., Harsha, P., Lachish, O., Matsliah, A.: Sound 3-Query PCPPs are Long. ACM Transactions on Computation Theory 1(2), 7:1–7:49 (2009); Preliminary version appeared in: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 686–697. Springer, Heidelberg (2008)Google Scholar
  16. [BSW12]
    Boneh, D., Segev, G., Waters, B.: Targeted malleability: Homomorphic encryption for restricted computations. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference (ITCS 1912), pp. 350–366 (2012)Google Scholar
  17. [BV07]
    Bogdanov, A., Viola, E.: Pseudorandom bits for polynomials. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), pp. 41–51 (2007)Google Scholar
  18. [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
  19. [CMT12]
    Cormode, G., Mitzenmacher, M., Thaler, J.: Practical verified computation with streaming interactive proofs. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference (ITCS 2012), pp. 90–112 (2012)Google Scholar
  20. [CRR11]
    Canetti, R., Riva, B., Rothblum, G.N.: Two 1-round protocols for delegation of computation. Cryptology ePrint Archive, Report 2011/518 (2011)Google Scholar
  21. [CTY11]
    Cormode, G., Thaler, J., Yi, K.: Verifying computations with streaming interactive proofs. Proceedings of the VLDB Endowment 5(1), 25–36 (2011)CrossRefGoogle Scholar
  22. [Dam92]
    Damgård, I.B.: 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
  23. [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
  24. [DFH12]
    Damgård, I. B., 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)CrossRefGoogle Scholar
  25. [DFK+92]
    Dwork, C., Feige, U., Kilian, J., Naor, M., Safra, M.: Low Communication 2-Prover Zero-Knowledge Proofs for NP. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 215–227. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  26. [DGW09]
    Dvir, Z., Gabizon, A., Wigderson, A.: Extractors and rank extractors for polynomial sources. Computational Complexity 18(1), 1–58 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [DLN+04]
    Dwork, C., Langberg, M., Naor, M., Nissim, K., Reingold, O.: Succinct NP proofs and spooky interactions (December 2004), www.openu.ac.il/home/mikel/papers/spooky.ps
  28. [EG85]
    El Gamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Transactions on Information Theory 31(4), 469–472 (1985)MathSciNetCrossRefGoogle Scholar
  29. [FG12]
    Fiore, D., Gennaro, R.: Publicly verifiable delegation of large polynomials and matrix computations, with applications. Cryptology ePrint Archive, Report 2012/281 (2012)Google Scholar
  30. [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
  31. [GGP10]
    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)CrossRefGoogle Scholar
  32. [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
  33. [GH98]
    Goldreich, O., Håstad, J.: On the complexity of interactive proofs with bounded communication. Information Processing Letters 67(4), 205–214 (1998)MathSciNetCrossRefGoogle Scholar
  34. [GKR08]
    Goldwasser, S., Kalai, Y.T., Rothblum, G.N.: Delegating computation: Interactive proofs for Muggles. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC 2008), pp. 113–122 (2008)Google Scholar
  35. [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
  36. [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 1985MathSciNetCrossRefzbMATHGoogle Scholar
  37. [GR05]
    Gabizon, A., Raz, R.: Deterministic extractors for affine sources over large fields. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 407–418 (2005)Google Scholar
  38. [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
  39. [GVW02]
    Goldreich, O., Vadhan, S., Wigderson, A.: On interactive proofs with a laconic prover. Computational Complexity 11(1/2), 1–53 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. [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
  41. [HK05]
    Håstad, J., Khot, S.: Query efficient PCPs with perfect completeness. Theory of Computing 1(1), 119–148 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  42. [HT98]
    Hada, S., Tanaka, T.: On the Existence of 3-Round Zero-Knowledge Protocols. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 408–423. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  43. [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
  44. [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
  45. [KR08]
    Kalai, Y.T., Raz, R.: Interactive PCP. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 536–547. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  46. [KR09]
    Kalai, Y.T., Raz, R.: Probabilistically Checkable Arguments. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 143–159. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  47. [Lip12]
    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)CrossRefGoogle Scholar
  48. [LPR10]
    Lyubashevsky, V., Peikert, C., Regev, O.: On Ideal Lattices and Learning with Errors over Rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  49. [Mei12]
    Meir, O.: Combinatorial PCPs with short proofs. In: Proceedings of the 26th Annual IEEE Conference on Computational Complexity (CCC 2012) (2012)Google Scholar
  50. [Mic00]
    Micali, S.: Computationally sound proofs. SIAM Journal on Computing 30(4), 1253–1298 (2000); Preliminary version appeared in FOCS 1994MathSciNetCrossRefzbMATHGoogle Scholar
  51. [Mie08]
    Mie, T.: Polylogarithmic two-round argument systems. Journal of Mathematical Cryptology 2(4), 343–363 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  52. [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
  53. [NN90]
    Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC 1990), pp. 213–223 (1990)Google Scholar
  54. [Pai99]
    Paillier, P.: Public-Key Cryptosystems Based on Composite Degree Residuosity Classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  55. [PQ12]
    Petit, C., Quisquater, J.-J.: On Polynomial Systems Arising from a Weil Descent. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 451–466. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  56. [SBV+12]
    Setty, S., Braun, B., Vu, V., Blumberg, A.J., Parno, B., Walfish, M.: Resolving the conflict between generality and plausibility in verified computation. Cryptology ePrint Archive, Report 2012/622 (2012)Google Scholar
  57. [SBW11]
    Setty, S., Blumberg, A.J., Walfish, M.: Toward practical and unconditional verification of remote computations. In: Proceedings of the 13th USENIX Conference on Hot Topics in Operating Systems (HotOS 2013), p. 29 (2011)Google Scholar
  58. [Sha92]
    Shamir, A.: IP = PSPACE. Journal of the ACM 39(4), 869–877 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  59. [SMBW12]
    Setty, S., McPherson, M., Blumberg, A.J., Walfish, M.: Making argument systems for outsourced computation practical (sometimes). In: Proceedings of the 2012 Network and Distributed System Security Symposium (NDSS 2012) (2012)Google Scholar
  60. [SVP+12]
    Setty, S., 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 21st USENIX Security Symposium (Security 2012) (2012)Google Scholar
  61. [Val77]
    Valiant, L.G.: Graph-Theoretic Arguments in Low-Level Complexity. In: Gruska, J. (ed.) MFCS 1977. LNCS, vol. 53, pp. 162–176. Springer, Heidelberg (1977)CrossRefGoogle Scholar
  62. [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 2013

Authors and Affiliations

  • Nir Bitansky
    • 1
  • Alessandro Chiesa
    • 2
  • Yuval Ishai
    • 3
  • Omer Paneth
    • 4
  • Rafail Ostrovsky
    • 5
  1. 1.Tel Aviv UniversityIsrael
  2. 2.MITUSA
  3. 3.TechnionIsrael
  4. 4.Boston UniversityUSA
  5. 5.UCLAUSA

Personalised recommendations