The notion of
 is formalized by requiring that for every malicious efficient verifier
, there exists an efficient simulator
that can reconstruct the view of
in a true interaction with the prover, in a way that is indistinguishable to
weakens this notions by switching the order of the quantifiers and only requires that for every distinguisher
, there exists a (potentially different) simulator
In this paper we consider various notions of zero-knowledge, and investigate whether their weak variants are equivalent to their strong variants. Although we show (under complexity assumption) that for the standard notion of zero-knowledge, its weak and strong counterparts are not equivalent, for meaningful variants of the standard notion, the weak and strong counterparts are indeed equivalent. Towards showing these equivalences, we introduce new non-black-box simulation techniques permitting us, for instance, to demonstrate that the classical 2-round graph non-isomorphism protocol of Goldreich-Micali-Wigderson  satisfies a “distributional” variant of zero-knowledge.
Our equivalence theorem has other applications beyond the notion of zero-knowledge. For instance, it directly implies the
dense model theorem
of Reingold et al (STOC ’08), and the leakage lemma of Gentry-Wichs (STOC ’11), and provides a modular and arguably simpler proof of these results (while at the same time recasting these result in the language of zero-knowledge).