In this paper we give nearly-tight reductions for modern implicitly authenticated Diffie-Hellman protocols in the style of the Signal and Noise protocols, which are extremely simple and efficient. Unlike previous approaches, the combination of nearly-tight proofs and efficient protocols enables the first real-world instantiations for which the parameters can be chosen in a theoretically sound manner. Our reductions have only a linear loss in the number of users, implying that our protocols are more efficient than the state of the art when instantiated with theoretically sound parameters. We also prove that our security proofs are optimal: a linear loss in the number of users is unavoidable for our protocols for a large and natural class of reductions.
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