Zero Knowledge and Soundness Are Symmetric

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4515)


We give a complexity-theoretic characterization of the class of problems in NP having zero-knowledge argument systems. This characterization is symmetric in its treatment of the zero knowledge and the soundness conditions, and thus we deduce that the class of problems in NP ∩ coNP having zero-knowledge arguments is closed under complement. Furthermore, we show that a problem in NP has a statistical zero-knowledge argument system if and only if its complement has a computational zero-knowledge proof system. What is novel about these results is that they are unconditional, i.e., do not rely on unproven complexity assumptions such as the existence of one-way functions.

Our characterization of zero-knowledge arguments also enables us to prove a variety of other unconditional results about the class of problems in NP having zero-knowledge arguments, such as equivalences between honest-verifier and malicious-verifier zero knowledge, private coins and public coins, inefficient provers and efficient provers, and non-black-box simulation and black-box simulation. Previously, such results were only known unconditionally for zero-knowledge proof systems, or under the assumption that one-way functions exist for zero-knowledge argument systems.


Commitment Scheme Argument System Knowledge Argument Promise Problem Zero Knowledge 
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.


  1. [AH]
    Aiello, W., Håstad, J.: Statistical zero-knowledge languages can be recognized in two rounds. J. Comput. Syst. Sci. 42(3), 327–345 (1991)zbMATHCrossRefGoogle Scholar
  2. [BCC]
    Brassard, G., Chaum, D., Crépeau, C.: Minimum disclosure proofs of knowledge. J. Comput. Syst. Sci. 37(2), 156–189 (1988)zbMATHCrossRefGoogle Scholar
  3. [CT]
    Cover, T.M., Thomas, J.A.: Elements of information theory, 2nd edn. Wiley Interscience, Hoboken (2006)zbMATHGoogle Scholar
  4. [ESY]
    Even, S., Selman, A., Yacobi, Y.: The complexity of promise problems with applications to public-key cryptography. Inform. Control 61(2), 159–173 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [GMR]
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [GMW]
    Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. J. ACM 38(1), 691–729 (1991)zbMATHMathSciNetGoogle Scholar
  7. [Gol1]
    Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)zbMATHCrossRefGoogle Scholar
  8. [Gol2]
    Goldreich, O.: On promise problems (a survey in memory of Shimon Even [1935-2004]). Technical Report TR05–018, ECCC (2005)Google Scholar
  9. [GSV]
    Goldreich, O., Sahai, A., Vadhan, S.: Honest verifier statistical zero-knowledge equals general statistical zero-knowledge. In: Proc. 30th STOC, pp. 399–408 (1998)Google Scholar
  10. [GV]
    Goldreich, O., Vadhan, S.: Comparing entropies in statistical zero-knowledge with applications to the structure of SZK. In: Proc. 14th Comput. Complex., pp. 54–73 (1999)Google Scholar
  11. [HILL]
    Håstad, J., Impagliazzo, R., Levin, L., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [HORV]
    Haitner, I., Ong, S., Reingold, O., Vadhan, S.: Instance-dependent commitments for statistical zero-knowledge proofs. In: preparation (March 2007)Google Scholar
  13. [HR]
    Haitner, I., Reingold, O.: Statistically-hiding commitment from any one-way function. Technical Report 2006/436, Cryptol. ePrint Arch (2006)Google Scholar
  14. [IL]
    Impagliazzo, R., Luby, M.: One-way functions are essential for complexity based cryptography. In: Proc. 30th FOCS, pp. 230–235 (1989)Google Scholar
  15. [Nao]
    Naor, M.: Bit commitment using pseudorandomness. J. Cryptol. 4(2), 151–158 (1991)zbMATHCrossRefGoogle Scholar
  16. [NOV]
    Nguyen, M., Ong, S., Vadhan, S.: Statistical zero-knowledge arguments for NP from any one-way function. In: Proc. 47th FOCS, pp. 3–14 (2006)Google Scholar
  17. [NV]
    Nguyen, M., Vadhan, S.: Zero knowledge with efficient provers. In: Proc. 38th STOC, pp. 287–295 (2006)Google Scholar
  18. [Oka]
    Okamoto, T.: On relationships between statistical zero-knowledge proofs. J. Comput. Syst. Sci. 60(1), 47–108 (2000)zbMATHCrossRefGoogle Scholar
  19. [Ost]
    Ostrovsky, R.: One-way functions, hard on average problems, and statistical zero-knowledge proofs. In: Proc. 6th Annual Structure in Complexity Theory Conference, pp. 133–138 (1991)Google Scholar
  20. [OV]
    Ong, S., Vadhan, S.: Zero knowledge and soundness are symmetric. Technical Report TR06-139, ECCC (2006)Google Scholar
  21. [OW]
    Ostrovsky, R., Wigderson, A.: One-way functions are essential for non-trivial zero-knowledge. In: Proc. 2nd Israel Symposium on Theory of Computing Systems, pp. 3–17 (1993)Google Scholar
  22. [PT]
    Petrank, E., Tardos, G.: On the knowledge complexity of NP. Combinatorica 22(1), 83–121 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  23. [SV]
    Sahai, A., Vadhan, S.: A complete problem for statistical zero knowledge. J. ACM 50(2), 196–249 (2003)MathSciNetGoogle Scholar
  24. [Vad]
    Vadhan, S.: An unconditional study of computational zero knowledge. SIAM J. Comput. 36(4), 1160–1214 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.School of Engineering and Applied SciencesHarvard UniversityCambridgeUSA

Personalised recommendations