Do distributed differentially-private protocols require oblivious transfer?

We study the cryptographic complexity of two-party differentially-private protocols for a large natural class of boolean functionalities. Information theoretically, McGregor et al. [FOCS 2010] and Goyal et al. [Crypto 2013] demonstrated several functionalities for which the maximal possible accuracy in the distributed setting is significantly lower than that in the client-server setting. Goyal et al. [Crypto 2013] further showed that "highly accurate" protocols in the distributed setting for any non-trivial functionality in fact imply the existence of one-way functions. However, it has remained an open problem to characterize the exact cryptographic complexity of this class. In particular, we know that semi-honest oblivious transfer helps obtain optimally accurate distributed differential privacy. But we do not know whether the reverse is true. We study the following question: Does the existence of optimally accurate distributed differentially private protocols for any class of functionalities imply the existence of oblivious transfer (or equivalently secure multi-party computation)? We resolve this question in the affirmative for the class of boolean functionalities that contain an XOR embedded on adjacent inputs. We give a reduction from oblivious transfer to: Any distributed optimally accurate ϵ-differentially private protocol with ϵ > 0 computing a functionality with a boolean XOR embedded on adjacent inputs. Any distributed non-optimally accurate ϵ-differentially private protocol with ϵ > 0, for a constant range of non-optimal accuracies and constant range of values of ϵ, computing a functionality with a boolean XOR embedded on adjacent inputs. Enroute to proving these results, we demonstrate a connection between optimally-accurate twoparty differentially-private protocols for functions with a boolean XOR embedded on adjacent inputs, and noisy channels, which were shown by Crépeau and Kilian [FOCS 1988] to be sufficient for oblivious transfer.