In this work, we give an explicit construction of PRPGs that achieve parameters that are

As a corollary, we obtain explicit PRPGs with seed length \(\tilde{O}(\log^{3/2}n)\) and error \( \epsilon = 1/ \mathrm{poly}(n)\) for ordered permutation branching programs of width \(w = \mathrm{poly}(n) \)with an arbitrary number of accept states. Previously, seed length \(o(\log^2n)\) was only known when both the width and the reciprocal of the error are subpolynomial, i.e. \(w= n^{o(1)} \) and \(\epsilon = 1/n^{o(1)}\)(Braverman, Rao, Raz, Yehudayoff, FOCS 2010 and SICOMP 2014).

The starting point for our results are the recent space-efficient algorithms for estimating random-walk probabilities in directed graphs by Ahmadenijad, Kelner, Murtagh, Peebles, Sidford, and Vadhan (FOCS 2020), which are based on spectral graph theory and space-efficient Laplacian solvers. We interpret these algorithms as giving PRPGs with large seed length, which we then derandomize to obtain our results. We also note that this approach gives a simpler proof of the original result of Braverman, Cohen, and Garg, as independently discovered by Cohen, Doron, Renard, Sberlo, and Ta-Shma (personal communication, January 2021). %B Electronic Colloquium on Computational Complexity (ECCC) %V 2021 %G eng %U https://eccc.weizmann.ac.il/report/2021/019/ %N 19