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Since, the only way our construction can get random bits is by taking them from the seed, we use pairwise independence to minimize the number of bits actually used. 6. We use the same construction for both extractors and pseudo-random generators, using our standard convention that boolean functions over Ð bits are equivalent Ð . Our construction is given as input the following ingreto strings of length Ò dients: ¾ ¯ The length of strings in the source distribution Ò. (Or alternatively for Ð where Ð is the number of inputs of the pseudo-random generators Ò hard function).
Thus, if we sample an element from an imperfect random source and interpret it as the truth table of a function with high probability it will not be computable by a small circuit with gates, and thus the NW-generator will fool . As this is true for any predicate , any Nisan-Wigderson style hardness versus randomness construction is an extractor! In fact, when viewing the Nisan-Wigderson generator in an information theoretic perspective, it is beneficial to replace the notion of hardness from computational to information theoretic and use Kolmogorov complexity instead of circuit complexity.
Subsequent work [SZ99, SSZ98, TS96, Zuc97, TS98, NTS99] used these extractors and more composition methods to significantly improve the parameters. (Some of these constructions produce weaker objects called ”dispersers” which are the ”one-sided” analogue of extractors). Important milestones achieved by this line of work is the construction of an extractor which works for all enÒ random bits to extract all the randomness out tropy levels and spends poly of the source [TS96], and an extractor which spends the ”asymptotically optimal” seed length Ç Ò ¯ to extract any constant fraction of the randomness in the source as long as the source contains Ò random bits, [Zuc97].