Abstract

We give an informal introduction to zero-knowledge proofs, and survey their role both in the interface between complexity theory and cryptography and as objects of complexity-theoretic study in their own right.

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References

  1. 1.
    Aaronson, S.: Quantum lower bound for the collision problem. In: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pp. 635–642. ACM, New York (2002)CrossRefGoogle Scholar
  2. 2.
    Arvind, V., Das, B.: Szk proofs for black-box group problems. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 6–17. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Aiello, W., Håstad, J.: Statistical zero-knowledge languages can be recognized in two rounds. Journal of Computer and System Sciences 42(3), 327–345 (1991) (Preliminary version in FOCS 1987)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. Journal of the ACM 45(1), 70–122 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation. SIAM Journal on Computing 37(1), 47–82(electronic) (2007)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Babai, L.: Trading group theory for randomness. In: Proceedings of the 17th Annual ACM Symposium on Theory of Computing (STOC), pp. 421–429 (1985)Google Scholar
  8. 8.
    Barak, B.: How to go beyond the black-box simulation barrier. In: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 106–115. IEEE Computer Society, Los Alamitos (2001)Google Scholar
  9. 9.
    Barak, B.: Constant-round coin-tossing with a man in the middle or realizing the shared random string model. In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 345–355 (2002)Google Scholar
  10. 10.
    Brassard, G., Chaum, D., Crépeau, C.: Minimum disclosure proofs of knowledge. Journal of Computer and System Sciences 37(2), 156–189 (1988)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Babai, L., Fortnow, L., Lund, C.: Nondeterministic exponential time has two-prover interactive protocols. Computational Complexity 1(1), 3–40 (1991)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Babai, L., Fortnow, L., Levin, L., Szegedy, M.: Checking computations in polylogarithmic time. In: STOC, pp. 21–31. ACM, New York (1991)Google Scholar
  13. 13.
    Barak, B., Goldreich, O.: Universal arguments and their applications. In: IEEE Conference on Computational Complexity, pp. 194–203 (2002)Google Scholar
  14. 14.
    Ben-Or, M., Gutfreund, D.: Trading help for interaction in statistical zero-knowledge proofs. Journal of Cryptology 16(2), 95–116 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ben-Or, M., Goldreich, O., Goldwasser, S., Håstad, J., Kilian, J., Micali, S., Rogaway, P.: Everything provable is provable in zero-knowledge. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 37–56. Springer, Heidelberg (1990)Google Scholar
  16. 16.
    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), pp. 113–131. ACM Press, New York (1988)Google Scholar
  17. 17.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 1–10 (1988)Google Scholar
  18. 18.
    Boppana, R.B., Håstad, J., Zachos, S.: Does co-NP have short interactive proofs? Information Processing Letters 25, 127–132 (1987)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Babai, L., Moran, S.: Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes. Journal of Computer and System Sciences 36, 254–276 (1988)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Barak, B., Sahai, A.: How to play almost any mental game over the net - concurrent composition via super-polynomial simulation. In: FOCS, pp. 543–552. IEEE Computer Society, Los Alamitos (2005)Google Scholar
  21. 21.
    Bogdanov, A., Trevisan, L.: On worst-case to average-case reductions for NP problems. SIAM Journal on Computing 36(4), 1119–1159(electronic) (2006)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 11–19 (1988)Google Scholar
  23. 23.
    Chen, X., Deng, X.: Settling the complexity of two-player nash equilibrium. In: FOCS, pp. 261–272. IEEE Computer Society, Los Alamitos (2006)Google Scholar
  24. 24.
    Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited. Journal of the ACM 51(4), 557–594(electronic) (2004)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    De Santis, A., Di Crescenzo, G., Persiano, G., Yung, M.: Image Density is complete for non-interactive-SZK. In: Automata, Languages and Programming, 25th International Colloquium, ICALP, pp. 784–795 (1998) (See also preliminary draft of full version, May 1999)Google Scholar
  26. 26.
    Damgård, I., Goldreich, O., Okamoto, T., Wigderson, A.: Honest verifier vs. dishonest verifier in public coin zero-knowledge proofs. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 325–338. Springer, Heidelberg (1995)Google Scholar
  27. 27.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. In: STOC 2006. Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 71–78. ACM, New York (2006)CrossRefGoogle Scholar
  28. 28.
    Damgård, I., Goldreich, O., Wigderson, A.: Hashing functions can simplify zero-knowledge protocol design (too). Technical Report RS-94–39, BRICS, November 1994. See Part 1 of [DGOW]Google Scholar
  29. 29.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Transactions on Information Theory 22(6), 644–654 (1976)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Di Crescenzo, G., Sakurai, K., Yung, M.: On zero-knowledge proofs: from membership to decision. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pp. 255–264. ACM Press, New York (2000)Google Scholar
  31. 31.
    Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43(2), 268–292 (1996)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Fürer, M., Goldreich, O., Mansour, Y., Sipser, M., Zachos, S.: On completeness and soundness in interactive proof systems. Advances in Computing Research 5, 429–442 (1989) (Preliminary version in FOCS 1987)Google Scholar
  33. 33.
    Fortnow, L.: The complexity of perfect zero-knowledge. Advances in Computing Research: Randomness and Computation 5, 327–343 (1989)Google Scholar
  34. 34.
    Fortnow, L., Rompel, J., Sipser, M.: On the power of multi-prover interactive protocols. Theoretical Computer Science 134(2), 545–557 (1994)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Goldreich, O., Goldwasser, S.: On the limits of non-approximability of lattice problems. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC), pp. 1–9 (1998)Google Scholar
  36. 36.
    Goldreich, O., Krawczyk, H.: On the composition of zero-knowledge proof systems. SIAM Journal on Computing 25(1), 169–192 (1996) (Preliminary version in ICALP 1990)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Goldreich, O., Kushilevitz, E.: A perfect zero-knowledge proof system for a problem equivalent to the discrete logarithm. Journal of Cryptology 6, 97–116 (1993)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM Journal on Computing 18(1), 186–208 (1989) (Preliminary version in STOC 1985)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM 38(1), 691–729 (1991) (Preliminary version in FOCS 1986)MathSciNetMATHGoogle Scholar
  40. 40.
    Goldreich, O., Oren, Y.: Definitions and properties of zero-knowledge proof systems. Journal of Cryptology 7(1), 1–32 (1994)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)MATHCrossRefGoogle Scholar
  42. 42.
    Goldwasser, S., Sipser, M.: Private coins versus public coins in interactive proof systems. Advances in Computing Research: Randomness and Computation 5, 73–90 (1989)Google Scholar
  43. 43.
    Goldreich, O., Sahai, A., Vadhan, S.: Honest verifier statistical zero-knowledge equals general statistical zero-knowledge. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC), pp. 399–408 (1998)Google Scholar
  44. 44.
    Goldreich, O., Sahai, A., Vadhan, S.: Can statistical zero-knowledge be made non-interactive? or On the relationship of SZK and NISZK. In: Wiener, M.J. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 467–484. Springer, Heidelberg (1999)Google Scholar
  45. 45.
    Gutfreund, D., Ta-Shma, A.: Worst-case to average-case reductions revisited. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX-RANDOM. LNCS, vol. 4627, pp. 569–583. Springer, Heidelberg (2007)Google Scholar
  46. 46.
    Goldreich, O., Vadhan, S.P.: Comparing entropies in statistical zero knowledge with applications to the structure of SZK. In: IEEE Conference on Computational Complexity, pp. 54–73. IEEE Computer Society, Los Alamitos (1999)Google Scholar
  47. 47.
    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM Journal on Computing 28(4), 1364–1396 (1999) Preliminary versions. In: STOC 1989 and STOC 1990Google Scholar
  48. 48.
    Haitner, I., Reingold, O.: Statistically-hiding commitment from any one-way function. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), 2007, New York (2007)Google Scholar
  49. 49.
    Impagliazzo, R., Luby, M.: One-way functions are essential for complexity based cryptography. In: Proceedings of the 30th Annual Symposium on Foundations of Computer Science (FOCS), pp. 230–235 (1989)Google Scholar
  50. 50.
    Impagliazzo, R., Yung, M.: Direct minimum-knowledge computations (extended abstract). In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 40–51. Springer, Heidelberg (1988)Google Scholar
  51. 51.
    Kilian, J.: A note on efficient zero-knowledge proofs and arguments (extended abstract). In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing (STOC), pp. 723–732 (1992)Google Scholar
  52. 52.
    Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. Journal of the ACM 39(4), 859–868 (1992)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Lindell, Y.: Protocols for bounded-concurrent secure two-party computation in the plain model. Chicago Journal of Theoretical Computer Science, pages Article 1, 50 (2006)Google Scholar
  54. 54.
    Luby, M., Micali, S., Rackoff, C.: How to simultaneously exchange a secret bit by flipping a symmetrically-biased coin. In: FOCS, pp. 11–21. IEEE, New York (1983)Google Scholar
  55. 55.
    Micali, S.: Computationally sound proofs. SIAM Journal on Computing 30(4), 1253–1298 (2000), Preliminary version in FOCS 1994Google Scholar
  56. 56.
    Micciancio, D., Vadhan, S.: Statistical zero-knowledge proofs with efficient provers: lattice problems and more. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 282–298. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  57. 57.
    Naor, M.: Bit commitment using pseudorandomness. Journal of Cryptology 4(2), 151–158 (1991); Preliminary version In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, Springer, Heidelberg (1990)Google Scholar
  58. 58.
    Nguyen, M.-H., Ong, S.J., Vadhan, S.: Statistical zero-knowledge arguments for NP from any one-way function. In: Proceedings of the 47th Annual Symposium on Foundations of Computer Science (FOCS), pp. 3–14. IEEE Computer Society, Los Alamitos, CA, USA (2006)Google Scholar
  59. 59.
    Naor, M., Ostrovsky, R., Venkatesan, R., Yung, M.: Perfect zero-knowledge arguments for NP using any one-way permutation. Journal of Cryptology 11(2), 87–108 (1998); Preliminary version In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, Springer, Heidelberg (1993)Google Scholar
  60. 60.
    Nguyen, M.-H., Vadhan, S.: Zero knowledge with efficient provers. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 287–295. ACM Press, New York (2006)Google Scholar
  61. 61.
    Okamoto, T.: On relationships between statistical zero-knowledge proofs. Journal of Computer and System Sciences, 60(1), 47–108 (2000), Preliminary version in STOC 1996Google Scholar
  62. 62.
    Ostrovsky, R.: One-way functions, hard on average problems, and statistical zero-knowledge proofs. In: Proceedings of the 6th Annual Structure in Complexity Theory Conference, pp. 133–138. IEEE Computer Society, Los Alamitos (1991)CrossRefGoogle Scholar
  63. 63.
    Ong, S.J., Vadhan, S.: Zero knowledge and soundness are symmetric. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, Springer, Heidelberg (2007)Google Scholar
  64. 64.
    Ostrovsky, R., Wigderson, A.: One-way functions are essential for non-trivial zero-knowledge. In: Proceedings of the 2nd Israel Symposium on Theory of Computing Systems, pp. 3–17. IEEE Computer Society, Los Alamitos (1993)CrossRefGoogle Scholar
  65. 65.
    Pass, R.: Bounded-concurrent secure multi-party computation with a dishonest majority. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 232–241. ACM, New York (2004)Google Scholar
  66. 66.
    Pass, R., Rosen, A.: Bounded-concurrent secure two-party computation in a constant number of rounds. In: FOCS, p. 404. IEEE Computer Society, Los Alamitos (2003)Google Scholar
  67. 67.
    Pass, R., Rosen, A.: New and improved constructions of non-malleable cryptographic protocols. In: STOC 2005: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 533–542. ACM, New York (2005)CrossRefGoogle Scholar
  68. 68.
    Pass, R., Shelat, A.: Unconditional characterizations of non-interactive zero-knowledge. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 118–134. Springer, Heidelberg (2005)Google Scholar
  69. 69.
    Shamir, A.: IP = PSPACE. Journal of the ACM 39(4), 869–877 (1992)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Sipser, M.: Introduction to the Theory of Computation, 2nd edn., Boston, MA, USA. Thomson Course Technology (2005)Google Scholar
  71. 71.
    Sahai, A., Vadhan, S.: A complete problem for statistical zero knowledge. Journal of the ACM, 50(2), 196–249 (2003), Preliminary version in FOCS 1997Google Scholar
  72. 72.
    Vadhan, S.: Probabilistic proof systems, part I — interactive & zero-knowledge proofs. In: Rudich, S., Wigderson, A. (eds.) Computational Complexity Theory. American Mathematical Society. IAS/Park City Mathematics Series, vol. 10 (2004)Google Scholar
  73. 73.
    Vadhan, S.P.: An unconditional study of computational zero knowledge. SIAM Journal on Computing, 36(4), 1160–1214 (2006). Preliminary version in FOCS 2004Google Scholar
  74. 74.
    Watrous, J.: Limits on the power of quantum statistical zero-knowledge. In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 459 (2002)Google Scholar
  75. 75.
    Wee, H.: Finding Pessiland. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 429–442. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  76. 76.
    Yao, A.C.-C.: How to generate and exchange secrets. In: FOCS. Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pp. 162–167. IEEE Computer Society, Los Alamitos (1986)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Salil Vadhan
    • 1
  1. 1.School of Engineering and Applied Science, Harvard University, Cambridge, MA 02138USA

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