TR00-084 Authors: Salil Vadhan, Amit Sahai

Publication: 6th November 2000 10:27

Downloads: 2775

Keywords:

We present the first complete problem for SZK, the class of (promise)

problems possessing statistical zero-knowledge proofs (against an

honest verifier). The problem, called STATISTICAL DIFFERENCE, is to

decide whether two efficiently samplable distributions are either

statistically close or far apart. This gives a new characterization

of SZK that makes no reference to interaction or zero knowledge.

We propose the use of complete problems to unify and extend the study

of statistical zero knowledge. To this end, we examine several

consequences of our Completeness Theorem and its proof, such as:

- A way to make every (honest-verifier) statistical zero-knowledge

proof very communication efficient, with the prover sending only one

bit to the verifier (to achieve soundness error 1/2).

- Simpler proofs of many of the previously known results about

statistical zero knowledge, such as the Fortnow and Aiello--Håstad

upper bounds on the complexity of SZK and Okamoto's result that SZK is

closed under complement.

- Strong closure properties of SZK which amount to constructing

statistical zero-knowledge proofs for complex assertions built out of

simpler assertions already shown to be in SZK.

- New results about the various measures of "knowledge complexity,"

including a collapse in the hierarchy corresponding to knowledge

complexity in the "hint" sense.

- Algorithms for manipulating the statistical difference between

efficiently samplable distributions, including transformations which

"polarize" and "reverse" the statistical relationship between a pair

of distributions.