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# hwsol5.5 - CSci 5403 Spring 2010 Homework 5.5 Ruoyu Sun The...

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CSci 5403, Spring 2010 Homework 5.5 Ruoyu Sun The ”set lower bound protocol” can be applied to the following problem: prove | S | ≥ 2 K versus | S | ≤ K . Claim 8.16.1 proves the following result: if S ⊂ { 0 , 1 } m , | S | ≤ 2 k 2 , then for p = | S | / 2 k , p Pr h R H m,k ,y R { 0 , 1 } k ( x S,h ( x ) = y ) 3 4 p. The original protocol( proving 2 K vs < K ) works because 4 3 < 2; now, since 4 3 < 1 . 4 < 2, the protocol also works for this problem.(Deﬁne q = Pr h R H m,k ,y R { 0 , 1 } k ( x S,h ( x ) = y ) ,p * = K/ 2 k . If | S | ≥ 2 K , then q 3 4 p (2) p * ≥ 1 . 05 p * ; if | S | ≤ K , then q p * . ) Similarly, the ”set upper bound” protocol can be applied to: prove | S | ≤ K versus | S | ≥ 2 K . Using these two results, the interactive proof system for our problem is de- scribed as follows.(We assume in both protocols, the completeness/soundness errors are both no more than 1 3 ). Note that if

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hwsol5.5 - CSci 5403 Spring 2010 Homework 5.5 Ruoyu Sun The...

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