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**1996/003**

**Title:**

On Monotone Function Closure of Statistical Zero-Knowledge

**Authors:**
Ronald Cramer, Ivan Damgaard

**Abstract:**

Assume we are given a language L with an honest verifier

perfect zero-knowledge proof system. Assume also that the proof system is an

Arthur-Merlin game with at most 3 moves. The class of such languages

includes all random self-reducible language, and also any language with a

perfect zero-knowledge non-interactive proof.

We show that such a language satisfies a certain closure property, namely

that languages constructed from L by applying certain monotone functions to

statements on membership in L have perfect zero-knowledge proof systems.

The new set of languages we can build includes L itself, but also for

example languages consisting of n words of which at least t are in L.

A similar closure property is shown to hold for the complement of L and for

statistical zero-knowledge. The property we need for the monotone functions used

to build the new languages is that there are efficient secret sharing schemes

for their associated access structures. This includes (but is not necessarily

limited to) all monotone functions with polynomial size monotone formulas.

**ePrint:**
https://eprint.iacr.org/1996/003

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