Santha-Vazirani源，确定性冷凝器和非常强的提取器
Santha-Vazirani sources, deterministic condensers and very strong extractors
摘要：
半随机源（也称为Santha-Vazirani（SV）源）的概念代表n位序列，其中第i位对先前i-1位的依赖性对于每个$i\in[n]$是有限的。如果第i位对剩余的n-1位的依赖性是有限的，则这是强SV源。即使是强SV源已知不允许（通用）确定性提取器，但它们具有种子提取器，因为它们的最小熵是$Ω(n)$。很明显，强SV源不仅仅是高最小熵源，这项工作探索了直觉。已知确定性冷凝器不存在一般的高最小熵源，我们构造任何常数$ε, δ\in (0,1)$的确定性冷凝器，映射n位来自一个强SV源的偏差最多$δ$到$Ω(n)$位的最小熵率至少$1-ε$。总之，我们观察到，确定性凝聚器是密切相关的非常强的提取器-建议加强强（种子）提取器的概念：特别是，我们的建设可以被视为非常强的提取器的家庭强Santha-Vazirani分布。非常强的提取器的概念要求输出保持不可预测，即使对于不仅知道种子值（如在强提取器的情况下），而且还知道提取器的输出对应于与前面的每个种子值相同的输入值（例如，在字典排序下）的人。非常强的提取器非常类似于原始的SV源概念，除了比特必须满足平均不可预测性要求。
英文摘要：
The notion of semi-random sources, also known as Santha-Vazirani (SV) sources, stands for a sequence of n bits, where the dependence of the i'th bit on the previous i-1 bits is limited for every $i\in[n]$. If the dependence of the i'th bit on the remaining n-1 bits is limited, then this is a strong SV-source. Even the strong SV-sources are known not to admit (universal) deterministic extractors, but they have seeded extractors, as their min-entropy is $Ω(n)$. It is intuitively obvious that strong SV-sources are more than just high-min-entropy sources, and this work explores the intuition. Deterministic condensers are known not to exist for general high-min-entropy sources, and we construct for any constants $ε, δ\in (0,1)$ a deterministic condenser that maps n bits coming from a strong SV-source with bias at most $δ$ to $Ω(n)$ bits of min-entropy rate at least $1-ε$. In conclusion we observe that deterministic condensers are closely related to very strong extractors - a proposed strengthening of the notion of strong (seeded) extractors: in particular, our constructions can be viewed as very strong extractors for the family of strong Santha-Vazirani distributions. The notion of very strong extractors requires that the output remains unpredictable even to someone who knows not only the seed value (as in the case of strong extractors), but also the extractor's outputs corresponding to the same input value with each of the preceding seed values (say, under the lexicographic ordering). Very strong extractors closely resemble the original notion of SV-sources, except that the bits must satisfy the unpredictability requirement only on average.
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