A Derandomized Switching Lemma and an Improved Derandomization of AC0

We describe a new pseudorandom generator for AC0. Our generator ε-fools circuits of depth d and size M and uses a seed of length Ŏ(log^d+4 M/ε). The previous best construction for $d \geq 3$ was due to Nisan, and had seed length Ŏ(log^2d+6 M/ε). A seed length of O(log^2d+Ω(1) M) is best possible given Nisan-type generators and the current state of circuit lower bounds. Seed length Ω(log^d M/ε) is a barrier for any pseudorandom generator construction given the current state of circuit lower bounds. For d=2, a pseudorandom generator of seed length Ŏ(log² M/ε) was known. Our generator is based on a "pseudorandom restriction'' generator which outputs restrictions that satisfy the conclusions of the Hastad Switching Lemma and that uses a seed of polylogarithmic length.