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Minimum-Knowledge Interactive Proofs for Decision Problems
Interactive communication of knowledge from the point of view of resource-bounded computational complexity is studied. Extending the work of Goldwasser, Micali, and Rackof [Proc. 17th Annual ACM Symposium on the Theory of Computing, 1985, pp. 291““304; .,18 (1989), pp. 186““208], the authors define a protocol transferring the result of any fixed computation to be minimum-knowledge if it communicates no additional knowledge to the recipient besides the intended computational result. It is proved that such protocols may be combined in a natural way so as to build more complex protocols. A protocol is introduced for two parties, a prover and a verifier, with the following properties:(1) Following the protocol, the prover gives to the verifier a proof of the value, 0 or 1, of a particular Boolean predicate, which is (assumed to be) hard for the verifier to compute. Such a deciding “interactive proof-system“ extends the interactive proof-systems of [op. cit.], which are used only to confirm that a certain predicate has value 1. (2) The protocol is minimum-knowledge. (3) The protocol is result-indistinguishable: an eavesdropper, overhearing an execution of the protocol, does not learn the value of the predicate that is proved. The value of the predicate is a cryptographically secure bit, shared by the two parties to the protocol. This security is achieved without the use of encryption functions, all messages being sent in the clear. These properties enable one to define a cryptosystem in which each user receives exactly the knowledge he is supposed to receive, and nothing more
Black-Box Computational Zero-Knowledge Proofs, Revisited: The Simulation-Extraction Paradigm
The concept of zero-knowledge proofs has been around for about 25 years. It has been redefined over and over to suit the special security requirements of protocols and systems. Common among all definitions is the requirement of the existence of some efficient ``device\u27\u27 simulating the view of the verifier (or the transcript of the protocol), such that the simulation is indistinguishable from the reality. The definitions differ in many respects, including the type and power of the devices, the order of quantifiers, the type of indistinguishability, and so on.
In this paper, we will scrutinize the definition of ``black-box computational\u27\u27 zero-knowledge, in which there exists one simulator \emph{for all} verifiers, the simulator has black-box access to the verifier, and the quality of simulation is such that the real and simulated views cannot be distinguished by polynomial tests (\emph{computational} indistinguishability).
Working in a theoretical model (the Random-Oracle Model), we show that the indistinguishability requirement is stated in a \emph{conceptually} inappropriate way: Present definitions allow the knowledge of the \emph{verifier} and \emph{distinguisher} to be independent, while the two entities are essentially coupled. Therefore, our main take on the problem will be \emph{conceptual} and \emph{semantic}, rather than \emph{literal}. We formalize the concept by introducing a ``knowledge extractor\u27\u27 into the model, which tries to extract the extra knowledge hard-coded into the distinguisher (if any), and then helps the simulator to construct the view of the verifier. The new paradigm is termed \emph{Simulation-Extraction Paradigm}, as opposed to the previous \emph{Simulation Paradigm}. We also provide an important application of the new formalization: Using the simulation-extraction paradigm, we construct one-round (i.e. two-move) zero-knowledge protocols of proving ``the computational ability to invert some trapdoor permutation\u27\u27 in the Random-Oracle Model. It is shown that the protocol cannot be proven zero-knowledge in the classical Simulation Paradigm. The proof of the zero-knowledge property in the new paradigm is interesting in that it does not require knowing the internal structure of the trapdoor permutation, or a polynomial-time reduction from it to another (e.g. an -complete) problem