Pseudorandomness for Read-Once, Constant-Depth Circuits

For Boolean functions computed by read-once, depth-D circuits with unbounded fan-in over the de Morgan basis, we present an explicit pseudorandom generator with seed length Õ(log n). The previous best seed length known for this model was Õ(log n), obtained by Trevisan and Xue (CCC ‘13 ) for all of AC (not just read-once). Our work makes use of Fourier analytic techniques for pseudorandomness introduced by Reingold, Steinke, and Vadhan (RANDOM ‘13 ) to show that the generator of Gopalan et al. (FOCS ‘12 ) fools read-once AC. To this end, we prove a new Fourier growth bound for read-once circuits, namely that for every F : {0, 1} → {0, 1} computed by a read-once, depth-D circuit, ∑ s⊆[n],|s|=k |F̂ [s]| ≤ O(logD−1 n), where F̂ denotes the Fourier transform of F over Z2 .