Recently, there has been a major theoretical breakthrough in program obfuscation, in the area of indistinguishability obfuscation.
Background: Indistinguishability obfuscation is a form of program obfuscation, with the following security property: if $C_1,C_2$ are two different circuits that both compute the same function, then $\text{Obf}(C_1)$ should be indistinguishable from $\text{Obf}(C_2)$, where $\text{Obf}(C)$ is the obfuscation of the circuit $C$. In other words, if there are multiple ways to implement a particular functionality, the obfuscation of an implementation doesn't reveal anything about the implementation choices you made, though it might reveal the input-output function completely.
The recent breakthrough shows how to achieve indistinguishability obfuscation of any circuit in NC1. This is a huge step forward in the theory and foundations of obfuscation. Unfortunately, the construction is highly inefficient -- the obfuscated circuit is hugely slower than the original -- so the scheme described in this breakthrough paper is totally impractical. However, often in cryptography if we focus in on special classes of functions, we can come up with specialized schemes that perform a lot better.
My question: What functions do we know how to obfuscate (in the sense of indistinguishability obfuscation) efficiently? Are there classes of functions for which we can provide indistinguishability obfuscation, where the obfuscated circuit/program is efficient, and if so, what are they? I'm fine even if the scheme relies on non-standard hardness assumptions, or even a construction where we have no proof of security but where it seems plausible that it might be secure.
