I tried to recover a password. When thinking of this I recognized that the problem "password recovery" is a very nice example of a NP problem. If you know the password it's very easy to verify it in polynomial time. BUT if you don't know the password you have to search the whole space of possible solutions which can be shown to take exponential time.
Now my question is: Doesn't this demonstrate that P != NP since "password recovery" is an element of NP that can be shown to require more than polynomial time to run?
The problem is not showing that password recovery is non-polynomial, since clearly it is -- you have to search an exponential space of answers.
Actually, "password-recovery" isn't really a description of a standard computational problem. It seems that, formally, password breaking algorithms take some sort of "oracle" that can answer whether a given string is the correct password. However, P and NP are defined in terms of Turing machines, which take strings as input.