Nisan-Wigderson Pseudorandom Generators for Circuits with Polynomial Threshold Gates Valentine Kabanets * Zhenjian Lu † January 20, 2018 Abstract We show how the classical Nisan-Wigderson (NW) generator [NW94] yields a nontrivial pseudorandom generator (PRG) for circuits with sublinearly many polynomial threshold func- tion (PTF) gates. For the special case of a single PTF of degree d on n inputs, our PRG for error ǫ has the seed size exp  O   d · log n · log log(n/ǫ)  ; this can give a super-polynomial stretch even for a sub-exponentially small error parameter ǫ = exp(-n γ ), for any γ = o(1). In contrast, the best known PRGs for PTFs of [MZ13, Kan12] cannot achieve such a small error, although they do have a much shorter seed size for any constant error ǫ. For the case of circuits with degree-d PTF gates on n inputs, our PRG can fool circuits with at most n α/d gates with error exp(-n α/d ) and seed length n O( √ α) , for any α> 1. While a similar NW PRG construction was observed by Lovett and Srinivasan [LS11] to work for the case of constant-depth (AC 0 ) circuits with few PTF gates, the application of the NW generator to the case of general (unbounded depth) circuits consisting of a sublinear number of PTF gates does not seem to have been explicitly stated before. We do so in this note. * School of Computing Science, Simon Fraser University, Burnaby, BC, Canada; kabanets@sfu.ca † School of Computing Science, Simon Fraser University, Burnaby, BC, Canada; zla54@sfu.ca ISSN 1433-8092