A Note on Black-Box Separations for Indistinguishability Obfuscation

Mahmoody et al. (TCC 2016-A) showed that basing indistinguishability obfuscation (IO) on a wide range of primitives in a black-box way is as hard as basing public-key cryptography on one-way functions. The list included any primitive P that could be realized relative to random trapdoor permutation or degree-O(1) graded encoding oracle models in a secure way against computationally unbounded polynomial-query attackers. In this work, relying on the recent result of Brakerski, Brzuska, and Fleischhacker (ePrint 2016/226) in which they ruled out statistically secure approximately correct IO, we show that there is no fully black-box constructions of IO from any of the primitives listed above, assuming the existence of one-way functions and NP 6⊆ coAM. At a technical level, we provide an alternative lemma to the Borel-Cantelli lemma that is useful for deriving black-box separations. In particular, using this lemma we show that attacks in idealized models that succeed with only a constant advantage over the trivial bound are indeed sufficient for deriving fully black-box separations from primitives that exist in such idealized models unconditionally.