Computational two-party correlation: A dichotomy for key-agreement protocols

Let π be an efficient two-party protocol that given security parameter κ, both parties output single bits X _κ and Y _κ , respectively. We are interested in how (X _κ , Y _κ ) "appears" to an efficient adversary that only views the transcript T _κ . We make the following contributions: • We develop new tools to argue about this loose notion, and show (modulo some caveats) that for every such protocol π, there exists an efficient simulator such that the following holds: on input T _κ , the simulator outputs a pair (X _κ ^' , Y _κ ) such that (X _κ ^' , Y _κ ^' , T _κ ) is (somewhat) computationally indistinguishable from (X _κ , Y _κ , T _κ ). • We use these tools to prove the following dichotomy theorem: every such protocol π is: - either uncorrelated - it is (somewhat) indistin-guishable from an efficient protocol whose parties interact to produce T _κ , but then choose their out-puts independently from some product distribution (that is determined in poly-time from T _κ ), - or, the protocol implies a key-agreement protocol (for infinitely many _κ '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. • We use the above dichotomy to make progress on open problems on minimal cryptographic assumptions required for differentially private mechanisms for the XOR function. • A subsequent work of Haitner et al. uses the above dichotomy to makes progress on a long-standing open question regarding the complexity of fair two-party coin-flipping protocols. We highlight the following ideas regarding our technique: • The simulator algorithm is obtained by a carefully designed "competition" between efficient algorithms attempting to forecast (X _κ , Y _κ )|T _κ=t . The winner is used to simulate the outputs of the protocol. • Our key-agreement protocol uses the simulation to reduce to an information theoretic setup, and is in some sense non-black box.