Cryptographic Obfuscation is a powerful cryptographic technique. My question is, can you do it partially?

Instead of obfuscating an algorithm completely, you can have an algorithm with a couple of constants, such that an outside observer can verify the algorithm, except for those constants.

One way to do this is to create an algorithm $O$ which takes an algorithm $A$ and constants $c$, and creates a document with obfuscated algorithm $A(c)$ along with a plaintext copy of $A$, and then signs it. So as not to leak the signing key, $O$ is also obfuscated. The problem is that this requires a trusted setup. Namely, $O$ could place a backdoor in the obfuscated copy of $A$. Since $O$ is obfuscated, there is no way to verify its source. It's a catch-22! Literally all the trust needs to be placed in the obfuscator of $O$.



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