If strong pseudorandom number generator exists then $BPP=P$ holds and if one way functions exists then $BPP\subseteq SUBEXP$ holds.
- What are the best statements we have proved that come close to converse to these statements?
My motivation is this $P=BPP$ and $NP\subseteq P/poly$ are consistent with what we know. However $MCSP\subseteq NP\implies$ $NP\subseteq P/poly$ gives no strong pseudorandom generators in $P/poly$ from Circuit Minimization Problem by Kabanets and Cai.
- So is it possible one way functions and pseudorandom number generators are not necessary for derandomization?
It might be that any argument that shows $P=BPP$ implies strong pseudorandom number generator exists should also show $NP\not\subseteq P/poly$ in that case $P=BPP$ and $NP\subseteq P/poly$ should not be consistent with what we know. However I cannot uncover a statement on this.
In an extreme it can very well be that $P=NP$ in which case we have derandomization and no one way functions and pseudorandom number generators. It is not at all clear why these should be considered essential for derandomization.
- Are there alternate proofs of $E$ needing exponential circuits for infinite input lengths leading to $P=BPP$ without going through these objects?