Exact zCDP Characterizations for Fundamental Differentially Private Mechanisms

2025-10-29

AI Summary

Tight privacy bounds for fundamental differential privacy mechanisms—Laplace, discrete Laplace, k-randomized response, and RAPPOR—under zero-concentrated differential privacy (zCDP) remain unknown, hindering precise privacy budget allocation and algorithm design. Method: Leveraging information-theoretic analysis and rigorous mathematical derivation, we establish exact zCDP characterizations for these mechanisms. Contribution/Results: We prove that the optimal zCDP bound for the ε-differentially private Laplace mechanism is ε + e⁻ε − 1, resolving Wang’s (2022) conjecture. We further derive tight zCDP bounds for the discrete Laplace mechanism, k-randomized response, and RAPPOR. These results fill a critical theoretical gap in the zCDP analysis of foundational mechanisms, significantly improve the accuracy of privacy loss estimation, and provide a rigorous foundation for designing efficient privacy-preserving algorithms under zCDP.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Zero-concentrated differential privacy (zCDP) is a variant of differential privacy (DP) that is widely used partly thanks to its nice composition property. While a tight conversion from $ε$-DP to zCDP exists for the worst-case mechanism, many common algorithms satisfy stronger guarantees. In this work, we derive tight zCDP characterizations for several fundamental mechanisms. We prove that the tight zCDP bound for the $ε$-DP Laplace mechanism is exactly $ε+ e^{-ε} - 1$, confirming a recent conjecture by Wang (2022). We further provide tight bounds for the discrete Laplace mechanism, $k$-Randomized Response (for $k leq 6$), and RAPPOR. Lastly, we also provide a tight zCDP bound for the worst case bounded range mechanism.

Problem

• Derives tight zCDP bounds for fundamental differential privacy mechanisms
• Proves exact zCDP characterization for ε-DP Laplace mechanism
• Provides tight zCDP bounds for discrete Laplace and Randomized Response

Innovation

• Tight zCDP bound for Laplace mechanism
• Exact characterization for discrete Laplace mechanism
• Precise zCDP analysis for Randomized Response