**Comparing Entropies in Statistical Zero-Knowledge with Applications to the Structure of SZK **

*Oded Goldreich and Salil Vadhan*

**Abstract: **We consider the following (promise) problem, denoted ED (for Entropy
Difference): The input is a pairs of circuits, and YES instances
(resp., NO instances) are such pairs in which the first (resp.,
second) circuit generates a distribution with noticeably higher
entropy.

On one hand we show that any language having a (honest-verifier) statistical zero-knowledge proof is Karp-reducible to ED. On the other hand, we present a public-coin (honest-verifier) statistical zero-knowledge proof for ED. Thus, we obtain an alternative proof of Okamoto's result by which HVSZK (i.e., Honest-Verifier Statistical Zero-Knowledge) equals public-coin HVSZK. The new proof is much simpler than the original one. The above also yields a trivial proof that HVSZK is closed under complementation (since ED easily reduces to its complement). Among the new results obtained is an equivalence of a weak notion of statistical zero-knowledge to the standard one.

**Category / Keywords: **Zero-Knowledge, Universal Hashing

**Publication Info: **Appeared in the THEORY OF CRYPTOGRAPHY LIBRARY and has been included in the ePrint Archive.

**Date: **received Dec 24th, 1998.

**Contact author: **salil at theory lcs mit edu

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