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Abstract: 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 tothe adversary. We simplify and improve an explicit construction of extractorsfor bit-fixing sources with sublogarithmic k due to Kamp and Zuckerman SICOMP2006, achieving error exponentially small in k rather than polynomially smallin k. Our main result is that when k is sublogarithmic in n, the short outputlength of this construction Olog k output bits is optimal for extractorscomputable by a large class of space-bounded streaming algorithms.Next, we show that a random function is an extractor for oblivious bit-fixingsources with high probability if and only if k is superlogarithmic in n,suggesting that our main result may apply more generally. In contrast, we showthat a random function is a static resp. adaptive exposure-resilient functionwith high probability even if k is as small as a constant resp. log log n. Noexplicit exposure-resilient functions achieving these parameters are known.



Author: Yakir Reshef, Salil Vadhan

Source: https://arxiv.org/



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