Error reduction for weighted PRGs against read once branching programs

Abstract

Weighted pseudorandom generators (WPRGs), introduced by Braverman, Cohen and Garg [5], are a generalization of pseudorandom generators (PRGs) in which arbitrary real weights are considered, rather than a probability mass. Braverman et al. constructed WPRGs against read once branching programs (ROBPs) with near-optimal dependence on the error parameter. Chattopadhyay and Liao [6] somewhat simplified the technically involved BCG construction, also obtaining some improvement in parameters. In this work we devise an error reduction procedure for PRGs against ROBPs. More precisely, our procedure transforms any PRG against length n width w ROBP with error 1/poly(n) having seed length s to a WPRG with seed length s + O(log w/ε · log log 1/ε). By instantiating our procedure with Nisan's PRG [17] we obtain a WPRG with seed length O(log n · log(nw) + log w/ε · log log 1/ε). This improves upon [5] and is incomparable with [6]. Our construction is significantly simpler on the technical side and is conceptually cleaner. Another advantage of our construction is its low space complexity O(log nw) + poly(log log 1/ε) which is logarithmic in n for interesting values of the error parameter ε. Previous constructions (like [5, 6]) specify the seed length but not the space complexity, though it is plausible they can also achieve such (or close) space complexity.