# Definitions and properties of zero-knowledge proof systems

- 994 Downloads
- 109 Citations

## Abstract

In this paper we investigate some properties of zero-knowledge proofs, a notion introduced by Goldwasser, Micali, and Rackoff. We introduce and classify two definitions of zero-knowledge: *auxiliary-input* zero-knowledge and *blackbox-simulation* zero-knowledge. We explain why auxiliary-input zero-knowledge is a definition more suitable for cryptographic applications than the original [GMR1] definition. In particular, we show that any protocol solely composed of subprotocols which are auxiliary-input zero-knowledge is itself auxiliary-input zero-knowledge. We show that blackbox-simulation zero-knowledge implies auxiliary-input zero-knowledge (which in turn implies the [GMR1] definition). We argue that all known zero-knowledge proofs are in fact blackbox-simulation zero-knowledge (i.e., we proved zero-knowledge using blackbox-simulation of the verifier). As a result, all known zero-knowledge proof systems are shown to be auxiliary-input zero-knowledge and can be used for cryptographic applications such as those in [GMW2].

We demonstrate the triviality of certain classes of zero-knowledge proof systems, in the sense that only languages in BPP have zero-knowledge proofs of these classes. In particular, we show that any language having a Las Vegas zero-knowledge proof system necessarily belongs to *RP*. We show that randomness of both the verifier and the prover, and nontriviality of the interaction are essential properties of (nontrivial) auxiliary-input zero-knowledge proofs.

## Key words

Zero-knowledge Computational complexity Computational indistinguishability Cryptographic composition of protocols## Preview

Unable to display preview. Download preview PDF.

## References

- [AH1]Aiello, W., and J. Hastad, Perfect Zero-Knowledge Languages Can Be Recognized in Two Rounds,
*Proc. 28th FOCS*, 1987, pp. 439–448.Google Scholar - [AH2]Aiello, W., and J. Hastad, Relativized Perfect Zero-Knowledge Is Not BPP,
*Inform. and Comput.*, Vol. 93, 1992, pp. 223–240.Google Scholar - [B]Babai, L., Trading Group Theory for Randomness,
*Proc. 17th STOC*, 1985, pp. 421–429.Google Scholar - [BCC]Brassard, G., D. Chaum, and C. Crepeau, Minimum Disclosure Proofs of Knowledge,
*J. Comput. System Sci.*, Vol. 37, No. 2, Oct. 1988, pp. 156–189.Google Scholar - [FS]Feige, U., and A. Shamir, Personal communication.Google Scholar
- [F]Fortnow, L., The Complexity of Perfect Zero-Knowledge,
*Proc. 19th STOC*, 1987, pp. 204–209.Google Scholar - [Gkg]Goldreich, O., S. Goldwasser, and S. Micali, How To Construct Random Functions,
*J. Assoc. Comput. Mach.*, Vol. 33, No. 4, 1986, pp. 792–807.Google Scholar - [GK]Goldreich, O., and H. Krawczyk, On the Composition of Zero-Knowledge Proof Systems,
*Proc. 17th ICALP*, Lecture Notes in Computer Science, Vol. 443, Springer-Verlag, Berlin, 1990, pp. 268–282.Google Scholar - [GMS]Goldreich, O., Y. Mansour, and M. Sipser, Interactive Proof Systems: Provers that Never Fail and Random Selection,
*Proc 28th FOCS*, 1987, pp. 449–461.Google Scholar - [GMW1]Goldreich, O., S. Micali, and A. Wigderson, Proofs that Yield Nothing but their Validity and a Methodology of Cryptographic Protocol Design,
*Proc. 27th FOCS*, 1986, pp. 174–187.Google Scholar - [GMW2]Goldreich, O., S. Micali, and A. Wigderson, How to Play any Mental Game or a Completeness Theorem for Protocols with Honest Majority,
*Proc. 19th STOC*, 1987, pp. 218–229.Google Scholar - [kg]Goldwasser, S., and S. Micali, Probabilistic Encryption,
*J. Comput. System Sci.*, Vol. 28, No. 2, 1984, pp. 270–299.Google Scholar - [GMR1]Goldwasser, S., S. Micali, and C. Rackoff, Knowledge Complexity of Interactive Proofs,
*Proc. 17th STOC*, 1985, pp. 291–304.Google Scholar - [GMR2]Goldwasser, S., S. Micali, and C. Rackoff, The Knowledge Complexity of Interactive Proof Systems,
*SIAM J. Comput.*, Vol. 18, No. 1, 1989, pp. 186–208.Google Scholar - [GS]Goldwasser, S., and M. Sipser, Arthur Merlin Games Versus Interactive Proof Systems,
*Proc. 18th STOC*, 1986, pp. 59–68.Google Scholar - [IY]Impagliazzo, R., and Yung, M., Direct Minimum-Knowledge Computations,
*Advances in Cryptology—Crypto 87*(proceedings), Lecture Notes in Computer Science, Vol. 293, Springer-Verlag, Berlin, 1987, pp. 40–51.Google Scholar - [O1]Oren, Y., Properties of Zero-Knowledge Proofs, M.Sc. Thesis, Computer Science Department, Technion, Haifa, Nov. 1987 (in Hebrew).Google Scholar
- [O2]Oren, Y., On the Cunning Power of Cheating Verifiers: Some Observations about Zero-Knowledge Proofs,
*Proc. 28th FOCS*, 1987, pp. 462–471.Google Scholar - [S]A. Shamir, IP = PSPACE,
*Proc. 31st FOCS*, 1990, pp. 11–15.Google Scholar - [TW]Tompa, M., and H. Woll, Random Self-Reducibility and Zero-Knowledge Interactive Proofs of Possession of Information,
*Proc. 28th FOCS*, 1987, pp. 472–482.Google Scholar - [Y]Yao, A. C., Theory and Applications of Trapdoor Functions,
*Proc. 23rd FOCS*, 1982, pp. 80–91.Google Scholar