What's the difference between the definitions of virtual black-box (VBB) obfuscation and indistinguishability obfuscation (iO)? Is there any presentation describing both? Furthermore, why is the notion of iO interesting (and sufficient in practical sense) given that it is a weaker notion than VBB?
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VBB is stronger, it states that accessing the (VBB obfuscated) representation is no better than looking at outputs and inputs.
VBB is also impossible in general case, meaning there exist functions that can not be obfuscated. The nonexistence of general VBB obfuscator doesn't mean VBB obfuscation is impossible as such. It merely means that such a theoretical obfuscator is not polynomial, or it can only work on a subclass of circuits.
Since VBB is stronger than iO, there are some tasks that can be achieved using VBB and cannot be using iO. One such thing is upgrading a symmetric cipher to a homomorphic cipher. You could prepare a circuit that decrypts some message (with hardcoced key), performs an operation and encrypts it again. Then you treat it with VBB.
iO will not work in such arrangement and you will be able to extract the hardcoded key from the obfuscated circuit. It could work however if you somehow obtained a VBB obfuscator magically.
haael
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