Skip to main content
SpringerLink
Log in

Probabilistic proof systems — A survey

  • Invited Talk
  • Conference paper
  • First Online:
STACS 97 (STACS 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1200))

Included in the following conference series:

  • 133 Accesses

Abstract

Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems — interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs — stressing the essential role of randomness in each of them.

Parts of the material presented in this survey have appeared in the Proceedings of the International Congress of Mathematicians 1994, Birkhäuser Verlag, Basel, 1995, pages 1395–1406.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. Aiello, M. Bellare and R. Venkatesan. Knowledge on the Average — Perfect, Statistical and Logarithmic. In 27th STOC, pages 469–478, 1995.

    Google Scholar 

  2. W. Aiello and J. Hastad. Perfect Zero-Knowledge Languages can be Recognized in Two Rounds. In 28th FOCS, pages 439–448, 1987.

    Google Scholar 

  3. S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy. Proof Verification and Intractability of Approximation Problems. In 33rd FOCS, pages 14–23, 1992.

    Google Scholar 

  4. S. Arora and S. Safra. Probabilistic Checkable Proofs: A New Characterization of NP. In 33rd FOCS, pages 1–13, 1992.

    Google Scholar 

  5. L. Babai. Trading Group Theory for Randomness. In 17th STOC, pages 421–420, 1985.

    Google Scholar 

  6. L. Babai, L. Fortnow, and C. Lund. Non-Deterministic Exponential Time has Two-Prover Interactive Protocols. Computational Complexity, Vol. 1, No. 1, pages 3–40, 1991. Preliminary version in 31st FOCS, 1990.

    Article  Google Scholar 

  7. L. Babai, L. Fortnow, L. Levin, and M. Szegedy. Checking Computations in Polylogarithmic Time. In 23rd STOC, pages 21–31, 1991.

    Google Scholar 

  8. L. Babai and S. Moran. Arthur-Merlin Games: A Randomized Proof System and a Hierarchy of Complexity Classes. JCSS, Vol. 36, pp. 254–276, 1988.

    Google Scholar 

  9. D. Beaver and J. Feigenbaum. Hiding Instances in Multioracle Queries. In 7th STACS, Springer Verlag, LNCS Vol. 415, pages 37–48, 1990.

    Google Scholar 

  10. M. Bellare and O. Goldreich. On Defining Proofs of Knowledge. In Crypto92, Springer Verlag, LNCS Vol. 740, pages 390–420, 1992.

    Google Scholar 

  11. M. Bellare, O. Goldreich and M. Sudan. Free Bits, PCPs and Non-Approximability — Towards Tight Results. In 36th FOCS, pages 422–431, 1995.

    Google Scholar 

  12. M. Bellare and S. Goldwasser. The Complexity of Decision versus Search. SIAM Journal on Computing, Vol. 23, pages 97–119, 1994.

    Article  Google Scholar 

  13. M. Ben-Or, O. Goldreich, S. Goldwasser, J. Håstad, J. Kilian, S. Micali and P. Rogaway. Everything Provable is Probable in Zero-Knowledge. In Crypto88, Springer Verlag, LNCS Vol. 403, pages 37–56, 1990

    Google Scholar 

  14. M. Ben-Or, S. Goldwasser, J. Kilian and A. Wigderson. Multi-Prover Interactive Proofs: How to Remove Intractability Assumptions. In 20th STOC, pages 113–131, 1988.

    Google Scholar 

  15. M. Blum, P. Feldman and S. Micali. Non-Interactive Zero-Knowledge and its Applications. In 20th STOC, pages 103–112, 1988.

    Google Scholar 

  16. M. Blum, M. Luby and R. Rubinfeld. Self-Testing/Correcting with Applications to Numerical Problems. JCSS, Vol. 47, No. 3, pages 549–595, 1993.

    Google Scholar 

  17. G. Brassard, D. Chaum and C. Crépeau. Minimum Disclosure Proofs of Knowledge. JCSS, pages 156–189, 1988. Extended abstract, by Brassard and Crépeau, in 27th FOCS, 1986.

    Google Scholar 

  18. C. Dwork, U. Feige, J. Kilian, M. Naor and S. Safra, Low Communication Perfect Zero Knowledge Two Provers Proof Systems. In Crypto92, Springer Verlag, LNCS Vol. 740, pages 215–227, 1992.

    Google Scholar 

  19. U. Feige. A Threshold of In n for Approximating Set Cover. In 28th STOC, pages 314–318, 1996.

    Google Scholar 

  20. U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy. Approximating Clique is almost NP-complete. In 32nd FOCS, pages 2–12, 1991.

    Google Scholar 

  21. U. Feige and J. Kilian. Zero knowledge and the chromatic number. In 11th IEEE Conference on Computational Complexity, pages 278–287, 1996.

    Google Scholar 

  22. U. Feige, D. Lapidot, and A. Shamir. Multiple non-interactive zero knowledge proofs based on a single random string. In 31st FOCS, pages 308–317, 1990.

    Google Scholar 

  23. L. Fortnow, The Complexity of Perfect Zero-Knowledge. In 19th STOC, pages 204–209, 1987.

    Google Scholar 

  24. L. Fortnow, J. Rompel and M. Sipser. On the Power of Multi-Prover Interactive Protocols. In Proc. 3rd IEEE Symp. on Structure in Complexity Theory, pages 156–161, 1988.

    Google Scholar 

  25. M. Furer, O. Goldreich, Y. Mansour, M. Sipser, and S. Zachos, “On Completeness and Soundness in Interactive Proof Systems”, Advances in Computing Research: a research annual, Vol. 5 (Randomness and Computation, S. Micali, ed.), pp. 429–442, 1989.

    Google Scholar 

  26. P. Gemmell, R. Lipton, R. Rubinfeld, M. Sudan, and A. Wigderson. Self-Testing/Correcting for Polynomials and for Approximate Functions. In 23th STOC, pages 32–42, 1991.

    Google Scholar 

  27. O. Goldreich and J. Håstad. On the Message Complexity of Interactive Proof Systems. Available as TR96-018 of ECCC, http://www.eccc.uni-trier.de/eccc/, 1996.

    Google Scholar 

  28. O. Goldreich, S. Micali and A. Wigderson. Proofs that Yield Nothing but their Validity or All Languages in NP Have Zero-Knowledge Proof Systems. JACM, Vol. 38, No. 1, pages 691–729, 1991. Extended abstract in 27th FOCS, 1986.

    Article  Google Scholar 

  29. O. Goldreich, S. Micali and A. Wigderson. How to Play any Mental Game or a Completeness Theorem for Protocols with Honest Majority. In 19th STOC, pages 218–229, 1987.

    Google Scholar 

  30. O. Goldreich and Y. Oren. Definitions and Properties of Zero-Knowledge Proof Systems. Journal of Cryptology, Vol. 7, No. 1, pages 1–32, 1994.

    Article  Google Scholar 

  31. O. Goldreich and E. Petrank. Quantifying Knowledge Complexity. In 32nd FOCS, pp. 59–68, 1991.

    Google Scholar 

  32. S. Goldwasser and S. Micali. Probabilistic Encryption. JCSS, Vol. 28, No. 2, pages 270–299, 1984. Extended abstract in 14th STOC, 1982.

    Google Scholar 

  33. S. Goldwasser, S. Micali and C. Rackoff. The Knowledge Complexity of Interactive Proof Systems. SIAM Journal on Computing, Vol. 18, pages 186–208, 1989. Extended abstract in 17th STOC, 1985.

    Article  Google Scholar 

  34. S. Goldwasser and M. Sipser. Private Coins versus Public Coins in Interactive Proof Systems. In 18th STOC, pages 59–68, 1986.

    Google Scholar 

  35. J. Håstad. Clique is hard to approximate within n 1-e. In 37th FOCS, pages 627–636, 1996.

    Google Scholar 

  36. J. Håstad. Getting optimal in-approximability results. Unpublish manuscript, June 1996. (Revised October 1996.)

    Google Scholar 

  37. J. Håstad, R. Impagliazzo, L.A. Levin and M. Luby. Construction of Pseudorandom Generator from any One-Way Function. Manuscript, 1993. See preliminary versions by Impagliazzo et. al. in 21st STOC and Håstad in 22nd STOC.

    Google Scholar 

  38. R. Impagliazzo and M. Yung. Direct Zero-Knowledge Computations. In Crypto87, Springer Verlag, LNCS Vol. 293, pages 40–51, 1987.

    Google Scholar 

  39. J. Kilian. A Note on Efficient Zero-Knowledge Proofs and Arguments. In 24th STOC, pages 723–732, 1992.

    Google Scholar 

  40. J. Kilian and E. Petrank. An Efficient Non-Interactive Zero-Knowledge Proof System for NP with General Assumptions. To appear in the Journal of Cryptography. Available as TR95-038 of ECCC, http://www.eccc.uni-trier.de/eccc/, 1995.

    Google Scholar 

  41. C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic Methods for Interactive Proof Systems. JACM, Vol. 39, No. 4, pages 859–868, 1992. Preliminary version in 31st FOCS, 1990.

    Article  Google Scholar 

  42. C. Lund and M. Yannakakis. On the Hardness of Approximating Minimization Problems, In 25th STOC, pages 286–293, 1993.

    Google Scholar 

  43. S. Micali. CS Proofs. In 35th FOCS, pages 436–453, 1994.

    Google Scholar 

  44. M. Naor. Bit Commitment using Pseudorandom Generators. Journal of Cryptology, Vol. 4, pages 151–158, 1991.

    Article  Google Scholar 

  45. R. Ostrovsky and A. Wigderson. One-Way Functions are essential for Non-Trivial Zero-Knowledge, In Proc. 2nd Israel Symp. on Theory of Computing and Systems (ISTCS93), IEEE Computer Society Press, pages 3–17, 1993.

    Google Scholar 

  46. C. H. Papadimitriou and M. Yannakakis. Optimization, Approximation, and Complexity Classes. In 20th STOC, pages 229–234, 1988.

    Google Scholar 

  47. E. Petrank and G. Tardos. On the Knowledge Complexity of NP. In 37th FOCS, pages 494–503, 1996.

    Google Scholar 

  48. R. Rubinfeld and M. Sudan. Robust Characterizations of Polynomials with Applications to Program Checking. SIAM J. of Computing, Vol. 25, No. 2, pages 252–271, 1996. Preliminary version in 3rd SODA, 1992.

    Article  Google Scholar 

  49. A. Shamir. IP=PSPACE. JACM, Vol. 39, No. 4, pages 869–877, 1992. Preliminary version in 31st FOCS, 1990.

    Article  Google Scholar 

  50. A.C. Yao. Theory and Application of Trapdoor Functions. In 23st FOCS, pages 80–91, 1982.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Oded Goldreich

Authors
  1. Oded Goldreich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rüdiger Reischuk Michel Morvan

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Goldreich, O. (1997). Probabilistic proof systems — A survey. In: Reischuk, R., Morvan, M. (eds) STACS 97. STACS 1997. Lecture Notes in Computer Science, vol 1200. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023492

Download citation

  • DOI: https://doi.org/10.1007/BFb0023492

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62616-9

  • Online ISBN: 978-3-540-68342-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation