Log-Seed Pseudorandom Generators via Iterated Restrictions

There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The "iterated restrictions" approach, pioneered by Ajtai and Wigderson [Ajtai and Wigderson, 1989], has provided PRGs with seed length polylog n or even Õ(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)? In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC⁰[⊕] formulas with seed length O(log n) + Õ(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(-Ω̃(log n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once 𝔽₂-polynomials) has seed length Θ(log n ⋅ (log log n)²) [Chin Ho Lee, 2019]. A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of ∏_{i=1}^m (1 + y_i). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [Gopalan and Yehudayoff, 2014] on elementary symmetric polynomials.