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We shall say that an obfuscator $\mathcal{O}$ is a reversible black box obfuscator if for each reversible program $P$ the obfuscated program $\mathcal{O}(P)$ is still reversible but does not reveal any more information than an oracle that computes $x\mapsto P(x)$ along with an oracle that computes $x\mapsto P^{-1}(x)$. It is well-known that black box obfuscators do not exist. Do reversible black box obfuscators exist?

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By "reversible program" I assume you mean an injective function.

The answer to your question is no, there is no black-box obfuscator for the set of injective functions.

The idea is to make an unobfuscatable function invertible using the Feistel trick. More formally, if $g: \{0,1\}^n \to \{0,1\}^n$ define $Feistel_g : \{0,1\}^{2n} \to \{0,1\}^{2n}$ as

$$Feistel_g(x,y) = (x, g(x) \oplus y).$$

$Feistel_g$ is invertible regardless of $g$ (and a similar thing works when the input/output lengths of $g$ are different).

Let $\mathcal{G}$ be the unobfuscatable function family from the Barak et al paper. Then define

$$Feistel_\mathcal{G} = \{ Feistel_g \mid g \in \mathcal{G} \}$$

Then this family consists of only invertible functions. Furthermore it is black-box unobfuscatable because black-box access to $g$ can be simulated given black-box access to $Feistel_g$ (and vice-versa). And oracle access to $(Feistel_g)^{-1}$ is redundant since $Feistel_g$ is its own inverse.

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