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3

This was actually the original proposal of Whit Diffie and Martin Hellman back in 1976: Alice puts $g^a$ in the telephone book and keeps $a$ secret, Bob does likewise with $g^b$ and $b$, and whenever Alice and Bob want to have a conversation they use $g^{ab}$ as a shared secret key. The main trouble is that you expose an oracle for $h \mapsto h^x$, where $...


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You can use any authenticated, secure cipher. AES-GCM is currently used a lot. You need a nonce per message to use AES-GCM or you can use one of the SIV variants, which calculate a nonce (Synthetic IV) assuming that the messages are all distinct. The question is more how to store the symmetric key than anything else, but that's not really a cryptography ...


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