exam2 - 2 there is exactly one linear function that fits...

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CSci 5403, Spring 2010 Exam 3 due: March 24, 2010 You may use the textbook and the class notes and example solutions, but no other sources to complete this exam. In particular, you may assume without proof that for every i 0 there exists a field F 2 i with 2 i elements and efficiently computable operations. (a) A family H of hash functions h : X Y is called t -wise independent if for every set of distinct x 1 , . . . , x t X and arbitrary ( y 1 , . . . , y t ) Y t , we have Pr h ∈H " t ^ i =1 ( h ( x i ) = y i ) # = 1 / | Y | t . Let Y be an arbitrary field, and give a construction of a t -wise independent hash family H (for arbitrary t ) with inputs and outputs from Y . Prove that your construction is t -wise independent. Hint : recall that the linear hash families used in previous homeworks (as well as the proof of Valiant-Vazirani) are pairwise independent because for every two points ( x 1 , y 1 ) and (
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Unformatted text preview: 2 ), there is exactly one linear function that fits both points. What sort of single-input function always uniquely fits t points? (b) Define the language u 2 sat = { φ | # φ = 2 } , that is, the the language of formulae with exactly 2 satisfying assignments. Use the existence of efficiently computable 3-wise independent hash functions to prove that NP ⊆ RP u 2 sat . (Note: this can be proven without using 3-wise independent hashes, but that is not the solution I’m looking for) Note: The constructions for both of these questions are fairly easy if you have been keeping up in class. What I am primarily looking for is a correct and detailed proof and analysis of the hash family (in part (a)) and oracle RP algorithm (in part (b)). 1...
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