Salil Vadhan

Abstract of PhD Thesis (MIT, 1999)

Title: A Study of Statistical Zero-Knowledge Proofs

Zero-knowledge interactive proofs, introduced by Goldwasser, Micali, and Rackoff, are fascinating constructs which enable one party (the "prover") to convince another party (the "verifier") of an assertion, with the property that the verifier learns nothing other than the fact that the assertion being proven is true. In addition to being powerful tools for constructing secure cryptographic protocols, zero-knowledge proofs yield rich classes of computational problems that are of both complexity-theoretic and cryptographic interest. This thesis is a detailed investigation of statistical zero-knowledge proofs, which are zero-knowledge proofs in which the condition that the verifier "learns nothing" is interpreted in a strong statistical sense. We begin by showing that the class SZK of problems possessing such proofs has two natural complete problems. These problems essentially amount to approximating the statistical difference or the difference in entropies between two "efficiently samplable" distributions. Thus, they give a new characterization of SZK which makes no reference to interaction or zero knowledge. They also simplify the study of statistical zero knowledge, as questions about the entire class SZK can be reduced to examining these two particular complete problems. Using these complete problems as tools, we proceed to answer a number of fundamental questions about zero-knowledge proofs, including:

• Transforming any statistical zero-knowledge proof against an honest verifier (i.e., a verifier that follows the specified protocol) into one which is zero knowledge even against cheating verifiers that deviate arbitrarily from the specified protocol. This transformation applies to public-coin computational zero-knowledge proofs as well.
• Constructing statistical zero-knowledge proofs for complex assertions built out of simpler assertions already shown to be in SZK. Via the complete problems, these closure properties translate to new methods for manipulating "efficiently samplable" distributions, which may be of independent interest.
• Obtaining simpler proofs of most previously known results about statistical zero knowledge, such as: Okamoto's result that SZK is closed under complement; the Fortnow and Aiello--Håstad upper bounds on the complexity of SZK; and Okamoto's result that every statistical zero-knowledge proof can be transformed into a public-coin one.

Thesis Supervisor: Shafi Goldwasser, RSA Professor of Computer Science

Submitted to the Department of Mathematics of MIT, Septemeber 1999.

Winner of the ACM Doctoral Dissertation Award, 2000.

Available: the thesis in PS and in PDF.

Related papers with O.G.: 1st, 2nd and 3rd.

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