The Coin Problem and Pseudorandomness for Branching Programs

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“…Existing constructions of pseudorandom generators [32,25] fail to reach this goal because they use seeds of length O(log 2 n). Despite significant effort, no improvement in the seed length has been achieved even when restricting attention to constant-width branching programs, although a recent, exciting line of works makes progress if the branching programs are constrained further [11,29,12]. In this work we show that the INW generator [25] can be adapted to use seed length O(log n · log log n) and produce an n-bit distribution in which each position cannot be predicted with advantage 1/ log n (given the previous positions) by poly-logarithmic width branching programs.…”

Randomness Buys Depth for Approximate Counting

“…The starting point is the following result which is essentially in [Ama09] and [BV10]. Later we use it with k := d − 2.…”

“…Instead, our starting points are recent works by Amano [Ama09] and by Brody and Verbin [BV10]. Using calculations similar to those in [Ajt83,ABO84], these works exhibit a deterministic circuit A of size poly(n) and depth d which solves the related problem of distinguishing the following two distributions with error 1/3: i.i.d.…”

“…For completeness in this section we give the proof of the following lemma which is essentially in [Ama09,BV10].…”

“…We thank Kord Eickmeyer for helpful initial discussions on this problem, and Joshua Brody and Elad Verbin for sending us a preliminary version of [BV10]. We are also grateful to Oded Goldreich and the anonymous referees for helpful comments on the write-up.…”