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The idea of using static key pairs is indeed that data in the key exchange gets signed. It doesn't matter too much what gets signed, as long as it the key exchange cannot be performed by an adversary. To verify signatures it is required that the public key of the signer needs to be trusted. This is what you are missing in the description. If the verifier ...


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There is a big question (that Meir Maor hints at) underlying your problem, namely: Must the algorithm work for a fixed group, or any group? Note that there exist groups for which DLOG (and things that reduce to DLOG) are easy, the most basic example is the group $(\mathbb{Z}/n\mathbb{Z}, +)$. Even among groups that have been used within cryptography, there ...


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