I apologize that this is a rather trivial question, but I haven't been able to find an answer anywhere. If the one-time pad is unconditionally secure and impossible to crack (with just ciphertext), why does this not imply $P \not= NP$? If not the one-time pad, what properties would a cryptographic system need to exhibit in order to imply $P \not= NP$?
A one time pad is secure regardless of complexity. When you have a ciphertext all plaintexts are equally likely and you have no way to verify a guess. Even an attacker which enumerates all possible keys will not learn anything he didn't know already. Thus it doesn't have anything to do with complexity.
A one way function on the other hand implies $P \ne NP$. Any function which is easy to compute in one direction but hard to find a pre image. Since finding a preimage of a polynomial time function is in $NP$.
Note it is possible that no one way functions exist and and still $P$ does not equal $NP$.
A one time pad is not a one way function. It is trivial to find a preimage We can find as many preimages as we want we can't tell them apart.