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Unformatted text preview: omorphism between H and either G1 or G2. Victor could just as well have generated this by himself. Because Victor can create a simulation of the protocol, it can be proven to be zero-knowledge. Hamiltonian Cycles
A variant of this example was first presented by Manuel Blum . Peggy knows a circular, continuous path along the lines of a graph that passes through each point exactly once. This is called a Hamiltonian cycle. Finding a Hamiltonian cycle is another hard problem. Peggy has this piece of information—she probably got it by creating the graph with a certain Hamiltonian cycle—and this is what she wants to convince Victor that she knows. Peggy knows the Hamiltonian cycle of a graph, G. Victor knows G, but not the Hamiltonian cycle. Peggy wants to prove to Victor that she knows this Hamiltonian cycle without revealing it. This is how she does it: (1) Peggy randomly permutes G. She moves the points around and changes their labels to make a new graph, H. Since G and H are topologically isomorphic (i.e., the same graph), if she knows the Hamiltonian cycle of G then she can easily find the Hamiltonian cycle of H. If she didn’t create H herself, determining the isomorphism between two graphs would be another hard problem; it could also take centuries of computer time. She then encrypts H to get H´. (This has to be a probabilistic encryption of each line in H, that is, an encrypted 0 or an encrypted 1 for each line in H.) (2) Peggy gives Victor a copy of H´. (3) Victor asks Peggy either to: (a) prove to him that H´ is an encryption of an isomorphic copy of G, or (b) show him a Hamiltonian cycle for H. (4) Peggy complies. She either: (a) proves that H´ is an encryption of an isomorphic copy of G by revealing the permutations and decrypting everything, without showing a Hamiltonian cycle for either G or H, or (b) shows a Hamiltonian cycle for H by decrypting only those lines that constitute a Hamiltonian cycle, without proving that G and H are topologically isomorphic. (5) Peggy and Victor repeat steps (1) through (4) n times. If Peggy is hones...
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- Fall '10