One-way function and $EXP$ All examples of one-way functions I have see till now are closely related to the assumption that $NP\neq P $ (or even weaker ones, such as $UP\neq P$), but why not considering the theorem $P\neq EXP$? 

In particular, why not building functions that in one way are in the class $FP$ and the inverse function computable in $FEXP$ (the class of functions extending the class $EXP$)?
If $P=NP$, then any one-way function can be inverted.  Any one-way function can be inverted using non-deterministic polynomial time.  Therefore, the assumption that $P\ne NP$ is necessary (but maybe not sufficient).