关于次对数熵的抽取器和曝光弹性函数
On Extractors and Exposure-Resilient Functions for Sublogarithmic Entropy
摘要：
我们研究确定性提取不经意的位固定源（又名。弹性函数）和具有小最小熵的确定弹性函数：在函数的n个输入比特中，k << n比特是均匀随机的并且对对手是未知的。我们简化并改进了Kamp和Zuckerman（SICOMP 2006）的亚对数k的比特固定源的提取器的显式构造，实现了k的指数小误差，而不是k的多项式小误差。我们的主要结果是，当k是sublogarithmic在n中，这种结构的短输出长度（O（log k）的输出位）是最佳的提取器可计算的一大类空间有界流算法。接下来，我们证明了随机函数是具有高概率的不经意比特固定源的提取器，当且仅当k在n中是超对数的，这表明我们的主要结果可以更普遍地应用。相反，我们证明了一个随机函数是静态的（分别是）。自适应）具有高概率的可靠弹性函数，即使k与常数一样小（分别log log n）。没有显式的可恢复的功能，实现这些参数是已知的。
英文摘要：
We study deterministic extractors for oblivious bit-fixing sources (a.k.a. resilient functions) and exposure-resilient functions with small min-entropy: of the function's n input bits, k << n bits are uniformly random and unknown to the adversary. We simplify and improve an explicit construction of extractors for bit-fixing sources with sublogarithmic k due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space-bounded streaming algorithms. Next, we show that a random function is an extractor for oblivious bit-fixing sources with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure-resilient function with high probability even if k is as small as a constant (resp. log log n). No explicit exposure-resilient functions achieving these parameters are known.
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