Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity

We present an explicit pseudorandom generator with seed length Õ((log n)w+1) for read-once, oblivious, width w branching programs that can read their input bits in any order. This improves upon the work of Impaggliazzo, Meka and Zuckerman (FOCS’12) where they required seed length n1/2+o(1). A central ingredient in our work is the following bound that we prove on the Fourier spectrum of branching programs. For any width w read-once, oblivious branching program B : {0, 1}n → {0, 1}, any k ∈ {1, . . . , n}, ∑ S⊆[n]:|S|=k |B̂(S)| ≤ O(log n). This settles a conjecture posed by Reingold, Steinke, and Vadhan (RANDOM’13). Our analysis crucially uses a notion of local monotonicity on the edge labeling of the branching program. We carry critical parts of our proof under the assumption of local monotonicity and show how to deduce our results for unrestricted branching programs. ∗[email protected]. Supported by NSF grant CCF-1412958 and the Simons Foundation. †[email protected]. Supported by a Simons Investigator Award (#409864, David Zuckerman) ‡[email protected]. Supported in part by NSF grant CCF-1749750. §[email protected]. Supported by a Motwani Postdoctoral Fellowship and by NSF grant CCF1749750. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 171 (2017)