Existing proofs that deduce BPP = P from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against nondeterministic circuits, we convert any randomized algorithm over inputs of length n running in time t n to a deterministic one running in time t 2+ for an arbitrarily small constant 0"> 0 . Such a slowdown is nearly optimal, as, under complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing).

Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size s with seed length (1 + ) log s , under the assumption that there exists a function f E that requires nondeterministic circuits of size at least 2 (1 − ) n , where = O ( ) . The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.