More on bounded independence plus noise: Pseudorandom generators for read-once polynomials

We construct pseudorandom generators with improved seed length for several classes of tests. First we consider the class of read-once polynomials over GF(2) in m variables. For error ε we obtain seed length Õ(log(m/ε)) log(1/ε), where Õ hides lower-order terms. This is optimal up to the factor Õ(log(1/ε)). The previous best seed length was polylogarithmic in m and 1/ε. Second we consider product tests f : {0, 1}m → C≤1. These tests are the product of k functions fi : {0, 1}n → C≤1, where the inputs of the fi are disjoint subsets of the m variables and C≤1 is the complex unit disk. Here we obtain seed length n · poly log(m/ε). This implies better generators for other classes of tests. If moreover the fi have outputs independent of n and k (e.g., {−1, 1}) then we obtain seed length Õ(n + log(k/ε)) log(1/ε). This is again optimal up to the factor Õ(log 1/ε), while the previous best seed length was ≥ √ k. A main component of our proofs is showing that these classes of tests are fooled by almost d-wise independent distributions perturbed with noise. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 167 (2017)