Recently, a fascinating candidate construction for indistinguishability obfuscation was proposed by Garg et al. (FOCS 2013). This has led to a flurry of discovery of intriguing constructions of primitives and protocols whose existence was not previously known (for instance, fully deniable encryption by Sahai and Waters, STOC 2014). Most of them explicitly rely on additional hardness assumptions, such as one-way functions.
Our goal is to get rid of this extra assumption. We cannot argue that indistinguishability obfuscation of all polynomial-time circuits implies the existence of one-way functions, since if $P = NP$, then program obfuscation (under the indistinguishability notion) is possible. Instead, the ultimate goal is to argue that if $P \neq NP$ and program obfuscation is possible, then one-way functions exist.
Our main result is that if $NP \not\subseteq ioBPP$ and there is an efficient (even imperfect) indistinguishability obfuscator, then there are one-way functions. In addition, we show that the existence of an indistinguishability obfuscator implies (unconditionally) the existence of SZK-arguments for $NP$. This, in turn, provides an alternative version of our main result, based on the assumption of hard-on-the average $NP$ problems. To get some of our results we need obfuscators for simple programs such as 3CNF formulas.
The paper: PDF.
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