Derandomization from Worst-Case Assumptions: Error-correcting codes and worst-case to average-case reductions

Definition: CCρ(f) ≥ s if s-sized circuits can compute f with probability at most ρ for a random input. That is, for every circuit family {Cn} with |Cn| ≤ s(n), Prx←R{0,1}n [Cn(x) = f(x)] < ρ. If CC1−1/(100n)(f) ≥ s we say that f is “mildly hard on the average” for s-sized circuits (every circuit will fail on a 1/(100n) fraction of the inputs) and if CC1(f) ≥ s we say that f is “worst-case hard” for s-sized circuits (every circuit will fail on at least one input). Assumption 1: ∃f ∈ E such that CC1−1/(100n)(f) ≥ 2n . That is, for every large enough n and 2n sized circuit C, Pr x←R{0,1} [C(x) = f(x)] ≤ 1− 1 100n