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.

翻译：零集中差分隐私（zCDP）是差分隐私（DP）的一种变体，因其良好的组合性质而被广泛使用。虽然最坏情况机制存在从ε-DP到zCDP的紧致转换，但许多常见算法满足更强的保证。本文推导了若干基础机制的紧致zCDP刻画。我们证明了ε-DP拉普拉斯机制的紧致zCDP界恰好为ε + e^{-ε} - 1，证实了Wang（2022）最近的猜想。进一步给出了离散拉普拉斯机制、k-随机响应（当k ≤ 6时）以及RAPPOR的紧致界。最后，我们还给出了最坏情况有界范围机制的紧致zCDP界。