Fooling Parity Tests with Parity Gates

Dan Gutfreund, Emanuele Viola

Abstract: We study the complexity of computing k-wise independent and ε-biased generators G : {0, 1} n → {0, 1} m . Specifically, we refer to the complexity of computing Gexplicitly, i.e. given x ∈ {0, 1} n and i ∈ {0, 1}log m computing the i-th output bit of G(x). [MNT90] show that constant depth circuits of size poly(n) cannot explicitly compute k-wise independent and ε-biased generators with seed length $n \leq 2^{\log^{o(1)} m}$ . In this work we show that DLOGTIME-uniform constant depth circuits of size poly(n) with parity gates can explicitly compute k-wise independent and ε-biased generators with seed length n roughly $\log m \ll 2^{\log^{o(1)} m}$ . In some cases the seed length of our generators is optimal up to constant factors, and in general up to polynomial factors. To obtain our results, we show a new construction of combinatorial designs, and we also show how to compute, in DLOGTIME-uniform AC 0, random walks of length log c n over certain expander graphs of size 2 n .