Exact zCDP Characterizations for Fundamental Differentially Private Mechanisms

TL;DR

This paper derives exact and tight zCDP characterizations for fundamental differential privacy mechanisms, improving understanding of their privacy guarantees and confirming recent conjectures.

Contribution

It provides the first exact zCDP bounds for several key mechanisms, including Laplace, discrete Laplace, RAPPOR, and bounded range mechanisms.

Findings

Exact zCDP bound for Laplace mechanism: $oldsymbol{ ext{epsilon} + e^{- ext{epsilon}} - 1}$

Tight bounds established for discrete Laplace, $k$-Randomized Response ($k extless= 6$), and RAPPOR

Confirmed a recent conjecture by Wang (2022) regarding the Laplace mechanism

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 $\epsilon$-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 $\epsilon$-DP Laplace mechanism is exactly $\epsilon + e^{-\epsilon} - 1$, 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.