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SIAMCOMP
2010

Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized

13 years 3 months ago
Uniform Direct Product Theorems: Simplified, Optimized, and Derandomized
The classical direct product theorem for circuits says that if a Boolean function f : {0, 1}n → {0, 1} is somewhat hard to compute on average by small circuits, then the corresponding k-wise direct product function fk(x1, . . . , xk) = (f(x1), . . . , f(xk)) (where each xi ∈ {0, 1}n) is significantly harder to compute on average by slightly smaller circuits. We prove a fully uniform version of the direct product theorem with information-theoretically optimal parameters, up to constant factors. Namely, we show that for given k and , there is an efficient randomized algorithm A with the following property. Given a circuit C that computes fk on at least fraction of inputs, the algorithm A outputs with probability at least 3/4 a list of O(1/ ) circuits such that at least one of the circuits on the list computes f on more than 1 − δ fraction of inputs, for δ = O((log 1/ )/k); moreover, each output circuit is an AC0 circuit (of size poly(n, k, log 1/δ, 1/ )), with oracle access to ...
Russell Impagliazzo, Ragesh Jaiswal, Valentine Kab
Added 30 Jan 2011
Updated 30 Jan 2011
Type Journal
Year 2010
Where SIAMCOMP
Authors Russell Impagliazzo, Ragesh Jaiswal, Valentine Kabanets, Avi Wigderson
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