hw8 - CSci 5403, Spring 2010 Homework 8 due: May 6, 2010 1....

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CSci 5403, Spring 2010 Homework 8 due: May 6, 2010 1. A calculation. Let X 1 ,...,X n be boolean independent random variables such that for all i , Pr[ X i = 1] = δ , and let X = n i =1 X i (mod 2). Prove that Pr[ X = 1] = 1 / 2+ 1 2 (1 - 2 δ ) n . 2. Not-so-pseudo. Show that for every k > 0 there exists a constant ± > 0 and a function G : { 0 , 1 } * → { 0 , 1 } * that is double-exponentially computable (i.e., computable in time 2 2 O ( n ) on inputs of length n ) with | G ( x ) | = ( | x | ) = 2 ± | x | , such that for all sufficiently large n and every circuit C of size at most 2 k±n , Pr[ C ( U ( n ) ) = 1] - Pr[ C ( G ( U n )) = 1] < 1 /n . 3. Complete Mess. Recall that E = S c> 0 DTIME (2 cn ). Give an example of a trivially E -complete problem. 4. Randomly Right. Let’s show that any E -complete problem can transformed into one that is randomly self-reducible . Recall that a language L is randomly self-reducible if there is a PPT algorithm M that, given O such that Pr x ∈{ 0 , 1
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