Differential Privacy with Imperfect Randomness

In this work we revisit the question of basing cryptography on imperfect randomness. Bosley and Dodis TCC'07 showed that if a source of randomness $$\mathcal {R}$$ is "good enough" to generate a secret key capable of encrypting k bits, then one can deterministically extract nearly k almost uniform bits from $$\mathcal {R}$$, suggesting that traditional privacy notions namely, indistinguishability of encryption requires an "extractable" source of randomness. Other, even stronger impossibility results are known for achieving privacy under specific "non-extractable" sources of randomness, such as the $$\gamma $$-Santha-Vazirani SV source, where each next bit has fresh entropy, but is allowed to have a small bias $$\gamma < 1$$ possibly depending on prior bits. We ask whether similar negative results also hold for a more recent notion of privacy called differential privacy Dwork et al., TCC'06, concentrating, in particular, on achieving differential privacy with the Santha-Vazirani source. We show that the answer is no. Specifically, we give a differentially private mechanism for approximating arbitrary "low sensitivity" functions that works even with randomness coming from a $$\gamma $$-Santha-Vazirani source, for any $$\gamma <1$$. This provides a somewhat surprising "separation" between traditional privacy and differential privacy with respect to imperfect randomness. Interestingly, the design of our mechanism is quite different from the traditional "additive-noise" mechanisms e.g., Laplace mechanism successfully utilized to achieve differential privacy with perfect randomness. Indeed, we show that any non-trivial "SV-robust" mechanism for our problem requires a demanding property called consistent sampling, which is strictly stronger than differential privacy, and cannot be satisfied by any additive-noise mechanism.