Low-end uniform hardness vs. randomness tradeoffs for AM
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Uniform derandomization from pathetic lower bounds
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Pseudorandom generators for combinatorial checkerboards
Computational Complexity
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The Nisan–Wigderson pseudo-random generator [19] was constructed to derandomize probabilistic algorithms under the assumption that there exist explicit functions which are hard for small circuits. We give the first explicit construction of a pseudo-random generator with asymptotically optimal seed length even when given a function which is hard for relatively small circuits. Generators with optimal seed length were previously known only assuming hardness for exponential size circuits [13,26]. We also give the first explicit construction of an extractor which uses asymptotically optimal seed length for random sources of arbitrary min-entropy. Our construction is the first to use the optimal seed length for sub-polynomial entropy levels. It builds on the fundamental connection between extractors and pseudo-random generators discovered by Trevisan [29], combined with the construction above. The key is a new analysis of the NW-generator [19]. We show that it fails to be pseudorandom only if a much harder function can be efficiently constructed from the given hard function. By repeatedly using this idea we get a new recursive generator, which may be viewed as a reduction from the general case of arbitrary hardness to the solved case of exponential hardness.