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Yi Wu (CMU) Joint work with Parikshit Gopalan (MSR SVC) Ryan ODonnell (CMU) David Zuckerman (UT Austin) Pseudorandom Generators for Halfspaces TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAAA

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Outline Introduction Pseudorandom Generators Halfspaces Pseudorandom Generators for Halfspaces Our Result Proof Conclusion 2

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Deterministic Algorithm Program InputOutput The algorithm deterministically outputs the correct result. 3

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Randomized Algorithm Program Input Output Random Bits. The algorithm outputs the correct result with high probability. 4

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Randomized Algorithms Primality testing ST-connectivity Order statistics Searching Polynomial and matrix identity verification Interactive proof systems Faster algorithms for linear programming Rounding linear program solutions to integer Minimum spanning trees shortest paths minimum cuts Counting and enumeration Matrix permanent Counting combinatorial structures Primality testing ST-connectivity Order statistics Searching Polynomial and matrix identity verification Interactive proof systems Faster algorithms for linear programming Rounding linear program solutions to integer Minimum spanning trees shortest paths minimum cuts Counting and enumeration Matrix permanent Counting combinatorial structures Primality testing ST-connectivity Order statistics Searching Polynomial and matrix identity verification Interactive proof systems Faster algorithms for linear programming Rounding linear program solutions to integer Minimum spanning trees shortest paths minimum cuts Counting and enumeration Matrix permanent Counting combinatorial structures Primality testing ST-connectivity Order statistics Searching Polynomial and matrix identity verification Interactive proof systems Faster algorithms for linear programming Rounding linear program solutions to integer Minimum spanning trees shortest paths minimum cuts Counting and enumeration Matrix permanent Counting combinatorial structures 5

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Is Randomness Necessary? Open Problem: Can we simulate every randomized polynomial time algorithm by a deterministic polynomial time algorithm (the BPP P cojecture)? Derandomization of randomized algorithms. Primality testing [AKS] ST-connectivity [Reingold] Quadratic residues [?] 6

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How to generate randomness? Question: How togenerate randomness for every randomized algorithm? Simpler Question: How to generate pseudorandomness for some class of programs? 7

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Pseudorandom Generator (PRG) PRG n random bit k