I need to prove that if John can ask Alice to sign any messages, John will eventually figure out secret key "d" and decrypt all Alice's messages. How do I go about proving that. Any hint will be appreciated.
What is known - public key (N,e),
Signature((N,e),msg) => msg^d (mod N)
One approach I'm thinking is find d from the definition of "e*d congruent 1 mod N".
Asking Alice to sign something arbitrarily is equivalent to a Chosen Ciphertext Attack because you signing is the same operation as decrypting ( $ m^d \mod n$). How save is that for RSA? that is discussed here: CC-Attack for RSA
Summary: you cannot get the $d$ directly (or there is no known fast way), so the answer to your original question would be "no". However, it could be used to decrypt any given Message (even without Alice noticing it).
Other links that explain that: here and here