exam3 - R S k , and r 1 ,...,r n R { , 1 } n . Output = E (...

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CSci 5403, Spring 2010 Exam 3 due: April 21, 2010 You may use the textbook and the class notes and example solutions, but no other sources to complete this exam. Graph Coloring. Recall that a graph G = ( V,E ) is k -colorable if there exists an assignment c : V [ k ] such that for all ( u,v ) E , c ( u ) 6 = c ( v ). It is not too hard to show that deciding whether G is k -colorable is NP -hard for any k > 2, and thus that there is no k +1 k approximation algorithm for coloring for any k > 2, unless P = NP . Prove that the following three-round protocol is a computational zero-knowledge interactive proof scheme for k -coloring: 1. P : let c be a k -coloring of G . Choose
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Unformatted text preview: R S k , and r 1 ,...,r n R { , 1 } n . Output = E ( ( c (1)); r 1 ) ,...,E ( ( c ( n )); r n ) 2. V : Choose at random e = ( u,v ) R E G . Output e . 3. P : Output h ( ( c ( u )) ,r u ) , ( ( c ( v )) ,r v ) i . 4. V : Accept i c u 6 = c v , E ( c u ; r u ) = u , and E ( c v ; r v ) = v . You may assume that E : [ k ] { , 1 } n { , 1 } * is 1-1 on [ k ] { , 1 } n ; E is eciently computable; and for all i,j [ k ], E ( i ; U n ) and E ( j ; U n ) are computationally indistinguish-able. 1...
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