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For the moment assume $g$ is a secret (uniformly random) generator, but that $p$ may be known to the adversary. Then given only $g^a, g^b$, the Diffie-Hellman key $g^{ab}$ is information-theoretically uniform (up to small statistical error), i.e., it cannot even be found by brute force because the adversary does not have enough information to determine it. ...

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