Pseudorandom generators for combinatorial checkerboards


We define a combinatorial checkerboard to be a function f: {1,⋯,m}d → {1,-1} of the form f(u1,⋯, ud) = Πi=1df(ui) for some functions fi: {1,⋯, m} → {1, -1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0,1} instead of {1, -1}. We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of small-bias generators, which correspond to the case m = 2. We construct a pseudorandom generator that ε-fools all combinatorial checkerboards with seed length O (log m + log d · log log d + log 3/2 1/ε). Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length O (log m +log2 d +log d ·log1/ε). Our seed length is better except when 1/ε ≥ d ω(log d). © 2011 IEEE.

Publication Title

Proceedings of the Annual IEEE Conference on Computational Complexity