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 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 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