Pseudorandom Generators for Read-Once Branching Programs, in any Order

A central question in derandomization is whether randomized logspace (RL) equals deterministic logspace (L). To show that RL=L, it suffices to construct explicit pseudorandom generators (PRGs) that fool polynomial-size read-once (oblivious) branching programs (roBPs). Starting with the work of Nisan, pseudorandom generators with seed-length $O(\log² n)$ were constructed. Unfortunately, improving on this seed-length in general has proven challenging and seems to require new ideas. A recent line of inquiry has suggested focusing on a particular limitation of the existing PRGs, which is that they only fool roBPs when the variables are read in a particular known order, such as $x1<\cdots<xn$. In comparison, existentially one can obtain logarithmic seed-length for fooling the set of polynomial-size roBPs that read the variables under any fixed unknown permutation $x{\pi(1)}<\cdots<x{\pi(n)}$. While recent works have established novel PRGs in this setting for subclasses of roBPs, there were no known $n^{o(1)}$ seed-length explicit PRGs for general polynomial-size roBPs in this setting. In this work, we follow the "bounded independence plus noise" paradigm of Haramaty, Lee and Viola, and give an improved analysis in the general roBP unknown-order setting. With this analysis we obtain an explicit PRG with seed-length $O(\log³ n)$ for polynomial-size roBPs reading their bits in an unknown order. Plugging in a recent Fourier tail bound of Chattopadhyay, Hatami, Reingold, and Tal, we can obtain a $\widetilde{O}(\log² n)$ seed-length when the roBP is of constant width.