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Pseudo-random generators for all hardnesses. (English) Zbl 1072.68129
Summary: Given a function $$f: \{0,1\}^{\log n}\to\{0,1\}$$ with circuit complexity $$s$$, we construct a pseudo-random generator that stretches a random seed of length $$O(\log n)$$ into a string of $$m=s^{\Omega(1)}$$ pseudo-random bits that fool circuits of size $$m$$. The construction works for any hardness $$s$$, giving an optimal hardness vs. randomness tradeoff with a direct and self-contained proof. A key element in our construction is an augmentation of the standard low-degree extension encoding that exploits the field structure of the underlying space in a new way.

##### MSC:
 68W20 Randomized algorithms 65C10 Random number generation in numerical analysis 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
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##### References:
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