6.841 Advanced Complexity Theory
May 11, 2009
Lecture 25
Lecturer: Madhu Sudan
Scribe: Rishi Gupta
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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 rockpaper
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 5050 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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 Spring '09
 MadhuSudan
 Cryptography, random bits, oneway function, Pseudorandomness, pseudorandom generator, pseudorandom generators

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