**Unprovable Security of Two-Message Zero Knowledge**

*Kai-Min Chung and Edward Lui and Mohammad Mahmoody and Rafael Pass*

**Abstract: **Goldreich and Oren (JoC'94) show that only trivial languages have 2-message zero-knowledge arguments. In this note we consider weaker, \emph{super-polynomial-time} simulation (SPS), notions of zero-knowledge. We present barriers to using black-box reductions for demonstrating soundness of 2-message protocols with efficient prover strategies satisfying SPS zero-knowledge. More precisely, we show that assuming the existence of $\poly(T(n))$-hard one-way functions, the following holds:

\begin{itemize} \item For sub-exponential (or smaller) $T(\cdot)$, \emph{polynomial-time} black-box reductions cannot be used to prove soundness of 2-message $T(\cdot)$-simulatable arguments based on any polynomial-time intractability assumption. This matches known 2-message quasi-polynomial-time simulatable arguments using a quasi-polynomial-time reduction (Pass'03), and 2-message exponential-time simulatable proofs using a polynomial-time reduction (Dwork-Naor'00, Pass'03).

\item $\poly(T(\cdot))$-time black-box reductions cannot be used to prove soundness of 2-message \emph{strong} $T(\cdot)$-simulatable (efficient prover) arguments based on any $\poly(T(\cdot))$-time intractability assumption; strong $T(\cdot)$-simulatability means that the output of the simulator is indistinguishable also for $\poly(T(\cdot))$-size circuits. This matches known 3-message strong quasi-polynomial-time simulatable proofs (Blum'86, Canetti et al'00). \end{itemize}

**Category / Keywords: **foundations / zero-knowledge, super-polynomial-time simulation, black-box lower bound, falsifiable assumptions, non-uniform

**Date: **received 19 Dec 2012

**Contact author: **chung at cs cornell edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20121219:163415 (All versions of this report)

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