## Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates (2005)

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Venue: | SIAM Journal on Computing |

Citations: | 25 - 13 self |

### BibTeX

@ARTICLE{Viola05pseudorandombits,

author = {Emanuele Viola},

title = {Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates},

journal = {SIAM Journal on Computing},

year = {2005},

volume = {36},

pages = {2007}

}

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### Abstract

We exhibit an explicitly computable ‘pseudorandom ’ generator stretching l bits into m(l) = l Ω(log l) bits that look random to constant-depth circuits of size m(l) with log m(l) arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS ’93) that achieves the same stretch but only fools circuits of depth 2 with one arbitrary symmetric gate at the top. Our generator fools a strictly richer class of circuits than Nisan’s generator for constant depth circuits (Combinatorica ’91) (but Nisan’s generator has a much bigger stretch). In particular, we conclude that every function computable by uniform poly(n)-size probabilistic constant depth circuits with O(log n) arbitrary symmetric gates is in TIME 2no(1)�. This seems to be the richest probabilistic circuit class known to admit a subexponential derandomization. Our generator is obtained by constructing an explicit function f: {0, 1} n → {0, 1} that is very hard on average for constant-depth circuits of size nɛ·log n with ɛ log 2 n arbitrary symmetric gates, and plugging it into the Nisan-Wigderson pseudorandom generator construction (FOCS ’88). The proof of the average-case hardness of this function is a modification of arguments by Razborov and Wigderson (IPL ’93), and Hansen and Miltersen (MFCS ’04), and combines H˚astad’s switching lemma (STOC ’86) with a multiparty communication complexity lower bound by Babai, Nisan and