[Solved] 2 Merlin Presents Arthur Graph G V E Claims G Contains Independent Set Size K Merlin Knows Q37263734
2. Merlin presents Arthur with a graph G-(V, E) and claims that G contains an independent set I of size k that only Merlin knows. Arthur wishes to verify this fact, however Merlin does not want to reveal the independent set to Arthur, so they agree to the following Proof Protocol: i) Merlin chooses a random permutation of vertices π ← perm(1, VB. and then com- putes G, r(G), such that if i E V, π(i) E V’, and edge (i,j) E E. then (n(i),r(j)) E E. Merlin sends G in a locked box to Arthur. Note: This is equivalent to computing an isomorphism of G ii) Arthur chooses b E [0, 1) uniformly randomly and tells his choice to Merlin iii) . If b-0, Merlin sends π, opens the locked box containing G. and reveals all the vertices in G. Arthur verifies that π(G) G. . If b-1, Merlin computes π(1), opens the box but only reveals the vertices u V, that are also in π(1) to Arthur, Arthur verifies that π(1) is an independent set of size k. Merlin and Arthur execute i)-iii) for k rounds until Arthur is confident that Merlin has the independent set, or until he discovers that Merlin has been cheating (a) Suppose Merlin does not have the purported independent set for G and has to cheat, what are his options? Hint 1: He needs one option for each b E {0, 1} Hint 2: Arthur doesn’t know how to solve graph non-isomorphism (b) For Cheating Merlin to fool Arthur, he has to correctly guess at the beginning of every round the value of a bit he has no information about. Show that the probability that Merlin can fool Arthur is at most. Justify your Arthur play for him to be at least 99.99% confident in Merlin’s independent set? n answer. How many rounds shoiu Show transcribed image text 2. Merlin presents Arthur with a graph G-(V, E) and claims that G contains an independent set I of size k that only Merlin knows. Arthur wishes to verify this fact, however Merlin does not want to reveal the independent set to Arthur, so they agree to the following Proof Protocol: i) Merlin chooses a random permutation of vertices π ← perm(1, VB. and then com- putes G, r(G), such that if i E V, π(i) E V’, and edge (i,j) E E. then (n(i),r(j)) E E. Merlin sends G in a locked box to Arthur. Note: This is equivalent to computing an isomorphism of G ii) Arthur chooses b E [0, 1) uniformly randomly and tells his choice to Merlin iii) . If b-0, Merlin sends π, opens the locked box containing G. and reveals all the vertices in G. Arthur verifies that π(G) G. . If b-1, Merlin computes π(1), opens the box but only reveals the vertices u V, that are also in π(1) to Arthur, Arthur verifies that π(1) is an independent set of size k. Merlin and Arthur execute i)-iii) for k rounds until Arthur is confident that Merlin has the independent set, or until he discovers that Merlin has been cheating (a) Suppose Merlin does not have the purported independent set for G and has to cheat, what are his options? Hint 1: He needs one option for each b E {0, 1} Hint 2: Arthur doesn’t know how to solve graph non-isomorphism (b) For Cheating Merlin to fool Arthur, he has to correctly guess at the beginning of every round the value of a bit he has no information about. Show that the probability that Merlin can fool Arthur is at most. Justify your Arthur play for him to be at least 99.99% confident in Merlin’s independent set? n answer. How many rounds shoiu
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