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 [Nis92], pseudorandom generators with seed-length O (log ² n) were constructed (see also [INW94], [GR14]). Unfortunately, improving on this seed-length in general has proven challenging and seems to require new ideas. A recent line of inquiry (e.g., [BV10], [GMR+12], [IMZ12], [RSV13], [SVW14], [HLV17], [LV17], [CHRT17]) has suggested focusing on a particular limitation of the existing PRGs ([Nis92], [INW94], [GR14]), which is that they only fool roBPs when the variables are read in a particular known order, such as x ₁ < · · · < xn. In comparison, existentially one can obtain logarithmic seed-length for fooling the set of polynomialsize roBPs that read the variables under any fixed unknown permutation x _π(1) < ··· < x _π(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 [HLV17], [LV17], and give an improved analysis in the general roBP unknown-order setting. With this analysis we obtain an explicit PRG with seed-length O(log ³ 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 [CHRT17], we can obtain a Õ(log ² n) seed-length when the roBP is of constant width.