Question / Idea: example to showcase "proof of having some secret"

[grasphoper]

One of the commonly advertised use cases for ZKProofs is the following:
I can prove to you that I have some secret S, where f(S) = X, without revealing the secret to you.

Most of the examples in the examples/ directory showcase "proving a result of some computation, so that the verifier doesn't have to redo the calculation again".

On the secrets front, there's the rsa example which comes close to what I have in mind but is different. In the rsa example, inside of the "program" which is our provable computation, we do roughly the following:

let pk = read_public_key();
let sig = read_signature();
let msg = read_signed_msg();
let valid_sig = RsaVerifySig(pk, sig, msg);
sp1_zkvm::io::commit(&valid_sig);

However, one can check whether an RSA signature is valid without using ZKProofs. I think it'd be interesting to showcase the ability of ZKProof to not reveal anything secret through proving roughly the following:

let public_key = read_public_key();
let private_key = read_private_key();
let valid_pair = public_key.Eq(RsaGeneratePublicKey(private_key));
sp1_zkvm::io::commit(&valid_pair);

This way we could prove that we have access to a private_key without generating any signatures through RSA.

Thanks for reading through my issue, let me know what you think!