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 epsilon-differentially private protocol with epsilon > 0 computing a functionality with a boolean XOR embedded on adjacent inputs. - Any distributed non-optimally accurate epsilon-differentially private protocol with epsilon > 0, for a constant range of non-optimal accuracies and constant range of values of epsilon, 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.
@InProceedings{goyal_et_al:LIPIcs.ICALP.2016.29, author = {Goyal, Vipul and Khurana, Dakshita and Mironov, Ilya and Pandey, Omkant and Sahai, Amit}, title = {{Do Distributed Differentially-Private Protocols Require Oblivious Transfer?}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {29:1--29:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.29}, URN = {urn:nbn:de:0030-drops-63080}, doi = {10.4230/LIPIcs.ICALP.2016.29}, annote = {Keywords: Oblivious Transfer, Distributed Differential Privacy, Noisy Channels, Weak Noisy Channels} }
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