Derandomization
The existence of certain strong pseudorandom number generators is conjectured by most experts of the field. This conjecture implies that randomness does not give additional computational power to polynomial time computation, that is, P = RP = BPP. Note that ordinary generators are not sufficient to show this result; any probabilistic algorithm implemented using a typical random number generator will always produce incorrect results on certain inputs irrespective of the seed (though these inputs might be rare).
László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson showed that unless EXPTIME collapses to MA, BPP is contained in i.o.-SUBEXP = . The class i.o.-SUBEXP, which stands for infinitely often SUBEXP, contains problems which have sub-exponential time algorithms for infinitely many input sizes. They also showed that P = BPP if the exponential-time hierarchy, which is defined in terms of the polynomial hierarchy and E as EPH, collapses to E; however, note that the exponential-time hierarchy is usually conjectured not to collapse.
Russell Impagliazzo and Avi Wigderson showed that if any problem in E, where, has circuit complexity then P = BPP.
Read more about this topic: BPP (complexity)