Computational Two-Party Correlation: A Dichotomy for Key-Agreement Protocols

Let $\pi$ be an efficient two-party protocol that given security parameter $\kappa$, both parties output single bits $X_\kappa$ and $Y_\kappa$, respectively. We are interested in how $(X_\kappa,Y_\kappa)$ "appears" to an efficient adversary that only views the transcript $T_\kappa$. We make the following contributions: $\bullet$ We develop new tools to argue about this loose notion and show (modulo some caveats) that for every such protocol \pi, there exists an efficient simulator such that the following holds: on input T_k, the simulator outputs a pair (X'_K ,Y'_K) such that $(X'_\kappa,Y'_\kappa,T_\kappa)$ is (somewhat) computationally indistinguishable from $(X_\kappa,Y_\kappa,T_\kappa)$. $\bullet$ We use these tools to prove the following dichotomy theorem: every such protocol $\pi$ is: -- either uncorrelated -- it is (somewhat) indistinguishable from an efficient protocol whose parties interact to produce T_k, but then choose their outputs independently from some product distribution (that is determined in poly-time from $T_\kappa$), -- or, the protocol implies a key-agreement protocol (for infinitely many $\kappa$'s). Uncorrelated protocols are uninteresting from a cryptographic viewpoint, as the correlation between outputs is (computationally) trivial. Our dichotomy shows that every protocol is either completely uninteresting or implies key-agreement. $\bullet$ We use the above dichotomy to make progress on open problems on minimal cryptographic assumptions required for differentially private mechanisms for the XOR function. $\bullet$ A subsequent work of Haitner et al. uses the above dichotomy to makes progress on a longstanding open question regarding the complexity of fair two-party coin-flipping protocols.