An Unconditional Study of Computational Zero Knowledge

Salil P. Vadhan


We prove a number of general theorems about ZK, the class of problems possessing (computational) zero-knowledge proofs. Our results are unconditional, in contrast to most previous works on ZK which rely on the assumption that one-way functions exist. We establish several new characterizations of ZK, and use these characterizations to prove results such as:

  • Honest-verifier ZK equals general ZK.
  • Public-coin ZK equals private-coin ZK.
  • ZK is closed under union (and more generally, "monotone formula closure").
  • ZK with imperfect completeness equals ZK with perfect completeness.
  • Any problem in [ZK intersect NP] can be proven in computational zero knowledge by a BPP^NP prover.
  • ZK with black-box simulators equals ZK with general, non-black-box simulators.

The above equalities refer to the resulting class of problems (and do not necessarily preserve other efficiency measures such as round complexity).

Our approach is to combine the conditional techniques previously used in the study of ZK with the unconditional techniques developed in the study of SZK, the class of problems possessing statistical zero-knowledge proofs. To enable this combination, we prove that every problem in ZK can be decomposed into a problem in SZK together with a set of instances from which a one-way function can be constructed.


  • In Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS `04),  pages 176-185, Rome, Italy, October 2004.  IEEE. [official IEEE page]
  • Full version, March 2005. [postscript][pdf]
  • Electronic Colloquium on Computational Complexity, Technical Report TR06-056, April 2006. [postscript][pdf][official ECCC page]
  • SIAM Journal on Computing 36 (4), pp. 1160-1214, 2006. Special Issue on Randomness and Complexity.  [official SICOMP page]

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