In the last fifteen years, there has been a steady stream of works combining differential privacy with various other cryptographic disciplines, particularly that of multi-party computation, yielding both practical and theoretical unification. As a part of that unification, due to the rich definitional nature of both fields, there have been many proposed definitions of differential privacy adapted to the given use cases and cryptographic tools at hand, resulting in computational and/or distributed versions of differential privacy. In this work, we offer a systemisation of such definitions, with a focus on definitions that are both computational and tailored for a multi-party setting. We order the definitions according to the distribution model and computational perspective and propose a viewpoint on when given definitions should be seen as instantiations of the same generalised notion. The ordering highlights a clear, and sometimes strict, hierarchy between the definitions, where utility (accuracy) can be traded for stronger privacy guarantees or lesser trust assumptions. Further, we survey theoretical results relating the definitions and extend some of them. We also discuss the state of well-known open questions and suggest new open problems to study. Finally, we consider aspects of the practical use of the different notions, hopefully giving guidance also to future applied work.

In this work we revisit the problem of using general-purpose MPC schemes to emulate the trusted dataholder in differential privacy (DP), to achieve the same accuracy but without the need to trust one single dataholder. In particular, we consider the two-party model where two computational parties (or dataholders), each with their own dataset, wish to compute a canonical DP mechanism on their combined data and to do so with active security. We start by remarking that available definitions of computational DP (CDP) for protocols are somewhat ill-suited for such a use-case, due to them either poorly capturing some strong security guarantees commonly given by general-purpose MPC protocols, or having too strict requirements in the sense that they need significant adjustment in order to be satisfiable by using common DP and MPC techniques. With this in mind, we propose a new version of simulation-based CDP, called SIM*-CDP, and prove it to be stronger than the IND-CDP and SIM-CDP and incomparable to SIM+-CDP. We demonstrate the usability of the SIM*-CDP definition by showing how to satisfy it by the use of an available distributed protocol for sampling truncated geometric noise. Further, we use the protocol to compute two-party inner-products with CDP and active security, and with accuracy equal to that of the central model, being the first to do so. Finally, we provide an open-sourced implementation and benchmark its practical performance. Our implementation generates a truncated geometric sample in between about 0.035 and 3.5 seconds (amortized), depending on network and parameter settings, comparing favourably to existing implementations.

Oblivious Pseudorandom Functions (OPRFs) are an elementary building block in cryptographic and privacy-preserving applications. While there are numerous pre-quantum secure OPRF constructions, it is unclear which of the proposed options for post-quantum secure constructions are practical for modern-day applications. In this work, we focus on isogeny group actions, as the associated low bandwidth leads to efficient constructions. We introduce OPUS, a novel Naor-Reingold-based OPRF from isogenies without oblivious transfer, and show efficient evaluations of the Naor-Reingold PRF using CSIDH and CSI-FiSh. Additionally, we analyze a previous proposal of a CSIDH-based OPRF and that the straightforward instantiation of the protocol leaks the server's private key. As a result, we propose mitigations to address those shortcomings, which require additional hardness assumptions. Our results report a very competitive protocol when combined with lattices for Oblivious Transfer.