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2011 26th Annual IEEE Conference on Computational Complexity
Pseudorandom Generators for Combinatorial Checkerboards
San Jose, California USA
June 08-June 11
ISBN: 978-0-7695-4411-3
We define a combinatorial checkerboard to be a function $f:\{1,\ldots,m\}^d\to\{1,-1\}$ of the form $f(u_1,\ldots,u_d)=\prod_{i=1}^df_i(u_i)$ for some functions $f_i:\{1,\ldots,m\}\to\{1,-1\}$. 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 $\epsilon$-fools all combinatorial checkerboards with seed length $O\bigl(\log m+\log d\cdot\log\log d+\log^{3/2}\frac{1}{\epsilon}\bigr)$. Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length $O\bigl(\log m+\log^2d+\log d\cdot\log\frac{1}{\epsilon}\bigr)$. Our seed length is better except when $\frac{1}{\epsilon}\geq d^{\omega(\log d)}$.
Index Terms:
pseudorandom generators, combinatorial checkerboards
Citation:
Thomas Watson, "Pseudorandom Generators for Combinatorial Checkerboards," ccc, pp.232-242, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
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