Is perfectly secret key exchange provably impossible? We know that perfect secrecy in encryption is possible (one-time pad). Now, the concept of key exchange like Diffie-Hellman is that we can establish a shared key without an interceptor knowing, and with no previous communication.

Obviously, for Diffie-Hellman as such, an interceptor can actually find the shared key; it would just take them ridiculously long. But I'm wondering, is it possible to make it impossible for the interceptor to find out the shared key?

I know that no such key exchange currently exists, but is it proven that it's impossible for such a scheme to exist? Or is there a chance it could be invented?
Lets start with a set of basic assumptions.


The communication channel transmits a sequence of symbols with each symbol taken from a finite set, there is no weird quantum stuff going on.
The attacker can obtain a complete record of the symbols that were transmitted.
The end systems are finite sized computers fed from a finite sized random number generator and run for a finite amount of time.
The purpose of the key exchange algorithm is to establish a shared secret.
No secrets were known by the end points prior to the start of the key exchange process.




I believe that in this case an attacker with unlimited computing power can always crack the key exchange. 

The attacker can simply enumerate all the possible outputs of the random number generator. Feed each one into the key exchange algorithm and compare the data that the algorithm generates with the data that was observed on the wire.

If the data exchanged on the wire matches then the shared secret must match.