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Average Bias and Polynomial Sources
We identify a new notion of pseudorandomness for randomness sources, which we
call the average bias. Given a distribution over , its average
bias is: . A source with average bias at most has
min-entropy at least , and so low average bias is a stronger condition than
high min-entropy. We observe that the inner product function is an extractor
for any source with average bias less than .
The notion of average bias especially makes sense for polynomial sources,
i.e., distributions sampled by low-degree -variate polynomials over
. For the well-studied case of affine sources, it is easy to see
that min-entropy is exactly equivalent to average bias of . We show
that for quadratic sources, min-entropy implies that the average bias is at
most . We use this relation to design dispersers for
separable quadratic sources with a min-entropy guarantee.Comment: We found out one of the main results has a much easier and direct
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