Resettable Statistical Zero Knowledge

Abstract

Two central notions of Zero Knowledge that provide strong, yet seemingly incomparable security guarantees against malicious verifiers are those of Statistical Zero Knowledge and Resettable Zero Knowledge. The current state of the art includes several feasibility and impossibility results regarding these two notions separately. However, the question of achieving Resettable Statistical Zero Knowledge (i.e., Resettable Zero Knowledge and Statistical Zero Knowledge simultaneously) for non-trivial languages remained open. In this paper, we show:

• Resettable Statistical Zero Knowledge with unbounded prover: under the assumption that sub-exponentially hard one-way functions exist, $\ensuremath{\mathcal{\text{r}SZK}}=\ensuremath{\mathcal{SZK}}$. In other words, every language that admits a Statistical Zero-Knowledge ($\ensuremath{\mathcal{SZK}}$) proof system also admits a Resettable Statistical Zero-Knowledge ($\ensuremath{\mathcal{\text{r}SZK}}$) proof system. (Further, the result can be re-stated unconditionally provided there exists a sub-exponentially hard language in $\mathcal{SZK}$). Moreover, under the assumption that (standard) one-way functions exist, all languages L such that the complement of L is random self reducible, admit a $\ensuremath{\mathcal{\text{r}SZK}}$; in other words: $\ensuremath{\mathcal{\text{co-}RSR}} \subseteq \ensuremath{\mathcal{\text{r}SZK}}$.

• Resettable Statistical Zero Knowledge with efficient prover: efficient-prover Resettable Statistical Zero-Knowledge proof systems exist for all languages that admit hash proof systems (e.g., QNR, QR, $\mathcal{DDH}$, DCR). Furthermore, for these languages we construct a two-round resettable statistical witness-indistinguishable argument system.

The round complexity of our proof systems is $\tilde O(\log \kappa)$, where κ is the security parameter, and all our simulators are black-box.

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