TR04-088 Authors: Emanuele Viola, Dan Gutfreund

Publication: 22nd October 2004 16:57

Downloads: 1372

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We study the complexity of computing $k$-wise independent and

$\epsilon$-biased generators $G : \{0,1\}^n \to \{0,1\}^m$.

Specifically, we refer to the complexity of computing $G$ \emph{explicitly}, i.e.

given $x \in \{0,1\}^n$ and $i \in \{0,1\}^{\log m}$, computing the $i$-th output bit of $G(x)$.

Mansour, Nisan and Tiwari (1990) show that constant depth circuits of size poly(n) (i.e. $AC^0$) cannot explicitly compute

$k$-wise independent and $\epsilon$-biased generators with seed length $n = 2^{\log^{o(1)} m}$.

In this work we show that DLOGTIME-uniform constant depth circuits

of size poly(n) \emph{with parity gates} can explicitly compute

$k$-wise independent and $\eps$-biased generators with seed length $n = (\poly) \log m$.

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 $2^n$.