Pseudorandom Bits for Constant Depth Circuits with Few Arbitrary Symmetric Gates

We exhibit an explicitly computable ‘pseudorandom’ generator stretching l bits into m(l) = lΩ(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 ...

• Type: Conference
• Year: 2005
• Where: COCO
• Authors: Emanuele Viola
• Keywords: Algorithms, Arbitrary Symmetric Gates, COCO 2005, Constant Depth Circuits, Nisan’s Generator