Amplifying Hardness: XOR and Hardcore Lemmas

Our goal today: Assumption 3 is still pretty strong in the sense that it says that no circuit can guess f(x) much better than the trivial 1/2. We will show that it is implied by the seemingly much weaker assumption that there’s some function f such that no circuit can compute f(x) with probability 1− 1/nc for some constant c > 0. Assumption 4: ∃f ∈ E such that CC1−n−c(f) ≥ 2n . That is, for every large enough n and 2n sized circuit C, Ex←R{0,1}n [RC,f (x)] ≤ 1− n −c