少比特破坏下最小熵的强链规则
Strong Chain Rules for Min-Entropy under Few Bits Spoiled
摘要：
众所周知，最小熵的概念不能满足$H(X,Y) = H(X|Y)+H(Y)$形式的\emph{chain rule}，即香农熵。这样的性质将有助于分析最小熵如何在较小的块中分裂。例如，当构造提取器和分散器时会出现这种问题。我们表明，任何序列的变量表现出一个非常强大的强大的块源结构（条件分布的块几乎是平坦的）时，我们\emph{spoil few correlated bits}。这意味着，以损坏位为条件，\emph{splitting-recombination properties}成立。特别是，我们有许多小熵一般不遵守的好性质，例如强链规则，“信息不会伤害”不等式，平均和最坏情况条件熵定义的等价性等。Quantitiment，对于任何序列$X_1,\ldots,X_t$的随机变量在一个字母表$\mathcal{X}$我们证明，当有条件的$m = t\cdot O( \log\log|\mathcal{X}| + \log\log(1/ε) + \log t)$位的辅助信息，所有条件分布的形式$X_i|X_{<i}$是$ε$ -接近接近平坦（只有一个常数因子）。这个论点是组合的（基于单纯形覆盖）。该结果可作为\emph{exhibiting block-source structures}的通用工具。我们证明了这一点，由于Nisan和Zuckermann（\emph{J. Computer and System Sciences, 1996}），这表明，从最小熵源采样块大致保持熵率的基本转换器。我们的界限意味着，只有通过简单的链式规则，$o(1)$（足够多的样本），这定性地满足了第一个更严格的分析，由于Vadhan（\emph{CRYPTO'03}），通过大偏差技术获得的这个问题的附加损失。
英文摘要：
It is well established that the notion of min-entropy fails to satisfy the \emph{chain rule} of the form $H(X,Y) = H(X|Y)+H(Y)$, known for Shannon Entropy. Such a property would help to analyze how min-entropy is split among smaller blocks. Problems of this kind arise for example when constructing extractors and dispersers. We show that any sequence of variables exhibits a very strong strong block-source structure (conditional distributions of blocks are nearly flat) when we \emph{spoil few correlated bits}. This implies, conditioned on the spoiled bits, that \emph{splitting-recombination properties} hold. In particular, we have many nice properties that min-entropy doesn't obey in general, for example strong chain rules, "information can't hurt" inequalities, equivalences of average and worst-case conditional entropy definitions and others. Quantitatively, for any sequence $X_1,\ldots,X_t$ of random variables over an alphabet $\mathcal{X}$ we prove that, when conditioned on $m = t\cdot O( \log\log|\mathcal{X}| + \log\log(1/ε) + \log t)$ bits of auxiliary information, all conditional distributions of the form $X_i|X_{<i}$ are $ε$-close to be nearly flat (only a constant factor away). The argument is combinatorial (based on simplex coverings). This result may be used as a generic tool for \emph{exhibiting block-source structures}. We demonstrate this by reproving the fundamental converter due to Nisan and Zuckermann (\emph{J. Computer and System Sciences, 1996}), which shows that sampling blocks from a min-entropy source roughly preserves the entropy rate. Our bound implies, only by straightforward chain rules, an additive loss of $o(1)$ (for sufficiently many samples), which qualitatively meets the first tighter analysis of this problem due to Vadhan (\emph{CRYPTO'03}), obtained by large deviation techniques.
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