EDIT: At a high level I am trying to create a scheme where Alice can prove to Bob that some external site has a certain piece of data (say her Bank account has balance > X) . She does not want to give Bob access to her account, but wants to prove that her bank website returns a certain result without showing it... basically I'm trying to build a system to authenticate and verify data from arbitrary HTTPS servers to someone that should not be granted access to the actual account or contents. I've come up with the following scheme:
$Alice$ requests some data over HTTPS from Google ($G$), she encrypts her cipher parameters $s$ using $G$'s public key $P_g$ and sends $P_g(s)$ to $G$. $G$ encrypts the returned data $d$ with the cipher $C$, using parameters $s$ returning $C(d,s)$ to $Alice$. Let's assume $d$ is just some number $x$.
$Alice$ then sends $Bob$ her cipher parameters encrypted with Googles public key: $P_g(s)$, and $C(d,s)$. Now $Bob$ can send $P_g(s)$ to $G$ and confirm that $G$ indeed returns $C(d,s)$. This can be done without $Bob$ knowing $s$.
Now $Alice$ would like to provide a proof to $Bob$ that $d=x$. For demonstrations sake let's assume that $C$ is RC4.
It seems like this could be done with zero knowledge proofs? Basically RC4 is a function being applied to a value and $Alice$ knows the exact structure of the function.
Can someone guide me towards how I would implement it for the above algorithms (RC4)