VDDP: Verifiable Distributed Differential Privacy under the Client-Server-Verifier Setup

Detailed Research Summary of Verifiable Distributed Differential Privacy (VDDP)

This research introduces Verifiable Distributed Differential Privacy (VDDP), a novel framework designed to address malicious behaviors within distributed differential privacy (DP) mechanisms operating in a client-server-verifier setup. The core innovation lies in providing rigorous end-to-end security and privacy guarantees against potential collusion among clients, servers, and verifiers by explicitly characterizing the connections and distinctions between VDDP and zero-knowledge proofs (ZKPs).

Core Framework and Formalization

The system is formalized through an Interactive Distributed Proof of Differential Privacy (I2DP) protocol. This framework incorporates randomness disentanglement (RD) to effectively resolve challenges arising from server collusion by strategically distributing the task of randomness generation between the verifier and the servers using a public-coin–private-coin model.

The security and privacy guarantees are established as ** (tau, epsilon) -verifiable distributed differential privacy (VDDP)** by ensuring several stringent conditions are met:

1. The underlying server functions must be (tau, epsilon(kappa), delta(kappa)) -DDP.

2. Commitment schemes must be hiding.

3. All sub-protocols (Cli and Ser) must be zero-knowledge (ZK).

The paper demonstrates that while Zero-Knowledge Proofs (ZKPs) are sufficient for achieving end-to-end DP, they are not strictly necessary for this level of security assurance within the VDDP framework.

Novel Instantiations and Efficiency Gains

The authors develop three distinct and highly efficient instantiations of the VDDP framework:

1. Verifiable Distributed Discrete Laplace Mechanism (VDDLM): This instantiation achieves a remarkable up to 400,000x improvement in proof generation efficiency compared to previous state-of-the-art verifiable differentially private mechanisms. Furthermore, it includes a tight privacy analysis that meticulously accounts for all additional privacy losses incurred due to numerical imprecisions arising from fixed-point arithmetic and truncation.

2. Verifiable Distributed Discrete Gaussian Mechanism (VDDGM): This is an extension of VDDLM, designed to maintain similar efficiency while incurring only limited overhead in real-world application scenarios.

3. Improved Verifiable Randomized Response (VRR) under Local DP: This represents a special case of VDDP, achieving up to a 5,000x reduction in communication costs and verifier overhead compared to prior solutions.

Key Contributions and Rigorous Analysis

The primary contributions of this work are threefold:

• Rigorous Framework Construction: The authors rigorously construct the VDDP framework for verifiable computations of DP mechanisms under the distributed client-server-verifier setup, successfully capturing potential collusive deviations from the protocol by clients and servers, as well as all privacy leakages across all participating parties (clients, servers, and verifiers).

• Development of Efficient Mechanisms: They develop VDDLM and VDDGM, coupled with a tight privacy analysis for VDDLM that incorporates numerical error losses.

• Overhead Reduction: The paper quantifies significant efficiency improvements: the expected L1 error for VDDLM is derived from parameters of the discrete Laplace distribution, and it is shown that the server function remains (nSer-1, epsilon, delta) -DDP when at least one server is fully honest.

Performance Metrics and Technical Findings

The research provides extensive empirical evidence supporting the efficiency claims:

• Proof Generation Speed: VDDLM is up to 400,000x faster in proof generation compared to VDBM under equivalent privacy costs.

• Communication and Verifier Overhead: Figure 9 illustrates a significant reduction across all perspectives. Specifically, for large problem dimensions (e.g., 1024) and small privacy budgets (epsilon=10-3), the server's and verifier's running times improve by 4 times 10 5 and 3 times 10 7 times, respectively, with communication cost reduced by a factor of 4 times 10 7.

• VRR Performance: The improved VRR solution achieves 0-completeness and (0, epsilon) -VDDP when ZK subroutines are included (Theorem 5.3). It also demonstrates constant and fixed per-client overheads (approximately 2.0 kB communication and 8 ms running time) under different probability space quantization configurations (Figure 12).

• Error Analysis: Theorem 4.7 provides an explicit expression for the expected L1 error in the modified discrete Laplace mechanism, linking it to parameters derived from the discrete Laplace noise distribution.

Open Questions

The paper concludes by identifying two critical open questions for future research:

1. Instantiations of VDDP that utilize non-zero-knowledge proofs while achieving asymptotic overhead reduction.

2. Alternative sampling methods for the discrete Gaussian mechanism that are compatible with existing or novel proof protocols.

1. Verifiable Distributed Differential Privacy (VDDP) framework allows for end-to-end security guarantees against potentially colluding adversaries of clients, servers, and verifiers by characterizing connections and distinctions between VDDP and zero-knowledge proofs (ZKPs). This enables the AI system to receive verifiable results even when servers deviate from the prescribed protocol.

2. The Verifiable Distributed Discrete Laplace Mechanism (VDDLM) can be used for counting queries, achieving up to a 400,000x improvement in proof generation efficiency compared to previous state-of-the-art verifiable differentially private mechanisms with only 0.1–0.2x error. This makes the AI system much faster when performing complex distributed counting operations under differential privacy constraints.

3. The Verifiable Distributed Discrete Gaussian Mechanism (VDDGM) is an extension of VDDLM that incurs limited overhead in real-world applications, allowing the AI system to utilize Gaussian noise for DP mechanisms while maintaining practicality. This enables the AI system to handle scenarios requiring Gaussian noise perturbations with manageable computational costs.

4. The improved solution to Verifiable Randomized Response (VRR) under local DP achieves up to a 5,000x reduction in communication costs and verifier overhead, allowing the AI system to perform randomized response tasks with significantly lower communication and verification burdens.

5. The I2DP protocol enables the system to verify the validity of client databases and the correctness of server computations, ensuring that an honest client Clij relies on the servers to engage in the protocol and verifying the correctness of each computation performed by the servers.

Although differential privacy (DP) is widely regarded as the de facto standard for data privacy, its implementation remains vulnerable to unfaithful execution by servers, particularly in distributed settings. In such cases, servers may sample noise from incorrect distributions or generate correlated noise while appearing to follow established protocols. This work addresses these malicious behaviours in a distributed client-server-verifier setup under Verifiable Distributed Differential Privacy (VDDP), a novel framework for the verifiable execution of distributed DP mechanisms. We systematically capture end-to-end security and privacy guarantees against potentially colluding adversarial behaviours of clients, servers, and verifiers by characterizing the connections and distinctions between VDDP and zero-knowledge proofs (ZKPs). As our primary instantiation of VDDP, the Verifiable Distributed Discrete Laplace Mechanism (VDDLM) achieves a proving-time speedup ranging from 50x to 4,000,000x over existing solutions, with larger gains in stronger-privacy regimes. Furthermore, the VDDLM sampling circuit achieves asymptotically lower overhead than prior constructions and is applicable to a broad range of cryptography-based DP applications.