ECDSA signature verifiable 1-way transformations Alice signs a message $m$ with her private key, yielding a signature ($r$,$s$).

I want to prove to someone else that I have this signature, but I don't want them to have the knowledge of what ($r$,$s$) is.

Is there a transformation I can apply to ($r$,$s$) to yield a new signature ($r'$,$s'$), such that someone who knows the original message and Alice's public key can see that it was derived from an original signature (without being able to reconstruct it)?

I'm thinking multiplying both the public key and signature components by some large number mod $n$. Am I on the right track or is this a fool's errand?
Use a zero-knowledge proof of knowledge (ZKPoK) of a value $(r,s)$ that is a valid signature.  For instance, you might be able to adapt existing ZKPoKs for proof of knowledge of a discrete logarithm to this problem.  Because it is zero-knowledge, you will know that it reveals nothing about $(r,s)$ and is not transferable.