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In the definition of indistinguishability obfuscation (iO), we have a probabilistic algorithm Obs that receives as input a circuit C, such that $i)$ the output Obs(C) is a circuit with the same functionality as C and $ii)$ for any $C_1$ and $C_2$ with the same functionality, we have that Obs($C_1$) is indistinguishable from Obs($C_2$).

Has a similar notion (and candidate construction) for a similar concept where the security holds for $C_1$ and $C_2$ are almost the same (i.e., they agree on all except a negligible fraction of the inputs, and it is computationally hard to find the points where they differ).

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  • $\begingroup$ Wouldn't "regular" iO imply this? Since $\text{Obs}$ needs to be efficient, in particular we can't use its output to distinguish between "almost-the-same" $C_1,C_2$, and so the outputs $\text{Obs}(C_1)$ and $\text{Obs}(C_2)$ must still be (computationally) indistinguishable as well? $\endgroup$
    – Clement C.
    Mar 5 '20 at 6:20
  • $\begingroup$ Not by the (text-book) definition of iO. The security definition only asks that of C_1 and C_2 have the same functionality, their obfuscation are indistinguishable and if C_1 is not implementing the same function as C_2, anything could happen. Answering your comment, I just realized that maybe what I am asking is equivalent for Virtual Black Box obfuscation for evasive functions (what is known to be impossible in the most general case) $\endgroup$
    – Alex Grilo
    Mar 6 '20 at 17:25
  • $\begingroup$ I understand this is not guaranteed by the definition itself, but my point is that it could be implied by it — since $\textrm{Obs}$ is assumed efficient, it could otherwise be used to "break" the computational hardness of distinguishing $C_1$ and $C_2$. Otherwise: take $C_1\neq C_2$ (different functionality) computationally hard to distinguish. Pass them through $\textrm{Obs}$. If you can distinguish $\textrm{Obs}(C_1)$ from $\textrm{Obs}(C_2)$, then you have distinguished $C_1$ from $C_2$ (which was assumed hard). So $\textrm{Obs}(C_1)$, $\textrm{Obs}(C_2)$ must also be hard to distinguish. $\endgroup$
    – Clement C.
    Mar 6 '20 at 18:16
  • $\begingroup$ (Unless I am botching something here, possibly two different notions of distinguishability?) $\endgroup$
    – Clement C.
    Mar 6 '20 at 18:19
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    $\begingroup$ I can post a longer answer later, but a keyword you should look up for is differing input obfuscation. $\endgroup$
    – Geoffroy Couteau
    Mar 7 '20 at 1:07
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I think the notion you're looking for is differing-inputs indistinguishability obfuscation(diO), or originally called extractability iO: https://eprint.iacr.org/2013/650.pdf

The definition of diO tells us: if there's a non-uniform PPT adversary that can distinguish the obfuscations of two circuits with "almost same" functionalities Obf($C_1$) and Obf($C_2$), then there exists a non-uniform PPT extractor that can extract an input(witness) $x$ where $C_1(x) \neq C_2(x)$. (the formal definition is on page 11)

If there are only polynomially many inputs where their outputs are different evaluated on $C_1, C_2$, then the standard iO definition also implies diO, using a binary search argument on the input space.

If there are negligible(in ratio) but exponentially many different outputs , then diO is a stronger assumption than standard iO. There are some negative results on the possibility of its existence in certain cases:

https://eprint.iacr.org/2016/162.pdf

https://eprint.iacr.org/2013/860.pdf

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