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 previousi − 1 bits is limited for every i ∈ [n]. If the dependence of the i’th bit on the remainingn − 1 bits is limited, then this is a strongSV-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 ∈ (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.

中文翻译：

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Santha-Vazirani源，确定性冷凝器和非常坚固的提取器

的概念半随机源，也称为桑沙-瓦齐拉尼（SV）源，代表的序列Ñ比特，其中第i个比特的对的依赖性以前我-  1个比特被限制为每个我∈[ n ]。如果在第i位的依赖性剩余Ñ -  1位是有限的，那么这是一个强SV -源。众所周知，即使是强大的SV源也不允许使用（通用的）确定性提取器，但是它们已经植入了种子提取器。，因为它们的最小熵是Ωn。从直觉上很明显，强大的SV源不仅是高最小熵源，而且这项工作探索了这种直觉。确定性冷凝器已知不用于一般高最小熵源存在，我们构建任何常数ε，δ＆Element;（0,1）确定性冷凝器该地图ñ位从强来SV -source与偏压至多δ至最小熵率的Ωn位至少为1- ε。总之，我们观察到确定性冷凝器与非常强大的提取器密切相关–建议加强强（种子）提取器的概念：尤其是，我们的构造可以看作是Santha-Vazirani强分布族的非常强大的提取器。非常强大的提取器的概念要求即使对于不仅知道种子值（例如强大的提取器）而且知道与每个先前种子值对应于相同输入值的提取器输出的人来说，输出仍然是不可预测的（例如，按照字典顺序）。非常强大的提取器与SV源的原始概念非常相似，不同之处在于，这些位仅必须平均地满足不可预测性要求。