Despite differential privacy (DP) often being considered the de facto standard for data privacy, its realization is vulnerable to unfaithful execution of its mechanisms by servers, especially in distributed settings. Specifically, servers may sample noise from incorrect distributions or generate correlated noise while appearing to follow established protocols. This work analyzes these malicious behaviours in a general differential privacy framework within a distributed client-server-verifier setup. To address these adversarial problems, we propose a novel definition called Verifiable Distributed Differential Privacy (VDDP) by incorporating additional verification mechanisms. We also explore the relationship between zero-knowledge proofs (ZKP) and DP, demonstrating that while ZKPs are sufficient for achieving DP under verifiability requirements, they are not necessary. Furthermore, we develop two novel and efficient mechanisms that satisfy VDDP: (1) the Verifiable Distributed Discrete Laplacian Mechanism (VDDLM), which offers up to a 400,000x improvement in proof generation efficiency with only 0.1-0.2x error compared to the previous state-of-the-art verifiable differentially private mechanism; (2) an improved solution to Verifiable Randomized Response (VRR) under local DP, a special case of VDDP, achieving up a reduction of up to 5,000x in communication costs and the verifier's overhead.

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