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# lect25 - 6.841 Advanced Complexity Theory Lecture 25...

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6.841 Advanced Complexity Theory May 11, 2009 Lecture 25 Lecturer: Madhu Sudan Scribe: Rishi Gupta Write your feedback for the course at https://sixweb.mit.edu/student/evaluate/6.841-s2009. It will help future students decide whether they should take the class. Motivation A few examples of where randomization (and derandomization) results are used: Algorithmic, for instance RP. This might be less exciting than we think though if BPP = P. Distributed Computing. A classic problem is: given n computers that are pairwise connected, each with a single bit. If all the bits are 0 they should agree that they all have 0, if they all have 1 they should agree they all have 1, otherwise they can do whatever. The communication links are arbitrarily speeds; some computers might even be asleep for an hour. We can’t solve this problem deterministically. Cryptography. Given that the inputs, outputs, and algorithm aren’t secret, randomness is essential to have any secrets at all. Game Theory. Equilibrium exists if and only if randomness exists. For instance, optimal rock-paper- scissors playing is dependent on access to random bits. Extracting Randomness Nature provides unpredictability. Physics at a small enough level is assumed to be random. However, as a practical matter, it’s hard to use this randomness to come up with unbiased, independent coins. Electrons are expensive to deal with; they don’t do things with exactly 50-50 probability, and it’s hard to do multiple independent trials on the same chip. Though intel does sell a costly randomness generating chip. So maybe we can start with a large number (say n 2 ) weakly random (biased, dependent) bits, and use those to generate n truly random bits. Von Neumann: If we start with a stream of random, independent bits, all biased with the same unknown

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