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تسجيل مستخدم جديدAverage Bias and Polynomial Sources
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نشر من قبل
Arnab Bhattacharyya
تاريخ النشر
2019
مجال البحث
الهندسة المعلوماتية
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English
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We identify a new notion of pseudorandomness for randomness sources, which we call the average bias. Given a distribution $Z$ over ${0,1}^n$, its average bias is: $b_{text{av}}(Z) =2^{-n} sum_{c in {0,1}^n} |mathbb{E}_{z sim Z}(-1)^{langle c, zrangle}|$. A source with average bias at most $2^{-k}$ has min-entropy at least $k$, 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 $2^{-n/2}$. The notion of average bias especially makes sense for polynomial sources, i.e., distributions sampled by low-degree $n$-variate polynomials over $mathbb{F}_2$. For the well-studied case of affine sources, it is easy to see that min-entropy $k$ is exactly equivalent to average bias of $2^{-k}$. We show that for quadratic sources, min-entropy $k$ implies that the average bias is at most $2^{-Omega(sqrt{k})}$. We use this relation to design dispersers for separable quadratic sources with a min-entropy guarantee.
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