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Yes, but typically not in any controlled way. Essentially you're asking whether two different encryption parameterisations map to the same ciphertext space and that is precisely what most cryptosystems do. Let's take AES-ECB as an example. Suppose that I have a key $k_0$ and a plaintext $P_0$ and a ciphertext $C=\mathrm{AES}_{\mathrm Enc}(k_0,P_0)$. Now I ...


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The definition of soundness for an interactive (zero-knowledge) proof says that no prover can convince a verifier of a false statement (i.e., one not in the language in question), except with tiny probability. In your “Bob in the middle” scenario the statement is true—the graph is 3-colorable—so there is no violation of soundness if “Bob” proves it by ...


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