# How exactly does derandomization of BPP imply lower bounds on NEXP

In their paper Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds, Kabanets and Impagliazzo prove that if $$BPP=P$$ then either $$(i) NEXP \nsubseteq P_{/poly}$$ or $$(ii)$$ permanent is not computable by poly-size arithmetic circuits .

Yet they claim that derandomizing Polynomial Identity Testing is essentially equivalent to proving arithmetic circuit lower bounds for NEXP. Where did the part about the permanent go ? My question is how come they say it's essentially equivalent.

My guess is that in some (morally speaking) way computing the permanent is in $$NEXP$$

• Computing permanent is in FP^PP which is in FNEXP.
– Mr.
Sep 24 at 22:35