Improved Pseudorandom Generators for Depth 2 Circuits

Abstract

We prove the existence of a poly(n,m)-time computable pseudorandom generator which “1/poly(n,m)-fools” DNFs with n variables and m terms, and has seed length O(log² nm ·loglognm). Previously, the best pseudorandom generator for depth-2 circuits had seed length O(log³ nm), and was due to Bazzi (FOCS 2007).

It follows from our proof that a $1/m^{\tilde O(\log mn)}$-biased distribution 1/poly(nm)-fools DNFs with m terms and n variables. For inverse polynomial distinguishing probability this is nearly tight because we show that for every m,δ there is a 1/m ^Ω(log1/δ)-biased distribution X and a DNF φ with m terms such that φ is not δ-fooled by X.

For the case of read-once DNFs, we show that seed length O(logmn ·log1/δ) suffices, which is an improvement for large δ.

It also follows from our proof that a 1/m ^O(log1/δ)-biased distribution δ-fools all read-once DNF with m terms. We show that this result too is nearly tight, by constructing a $1/m^{\tilde \Omega(\log 1/\delta)}$-biased distribution that does not δ-fool a certain m-term read-once DNF.