Notes for Lecture 16 1 Proof of the Impagliazzo-widgerson Theorem 1.1 Recap

We are near the end of our proof of the Impagliazzo-Widgerson result that every exponential time problem has a polynomial-time transformation to an equivalent exponential-time problem which is hard on average. To recap from last time: Suppose L is a language decidable in time 2O(n). Let Ln be the characteristic function of L on inputs of length n, Ln : {0, 1} n → {0, 1}. Choose an arbitrarily small constant γ > 0, a field F such that |F| = 2, and a set H ⊆ F with H = 2. We can define the polynomial p : F → F with degree(p) ≤ |H|m = 2γn+O(1), m = 1 γ , that agrees with Ln on H m ≈ {0, 1}. There exist absolute constants c and c′ such that if c is a circuit of size ≤ 2 that computes p, ie,