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Some observations, though I would recommend against inventing your own protocol, at least for real world use: When a new Chat group is formed, a random AES Key is generated and encrypted using each users public key. This means the protocol lacks forward secrecy. Anyone who compromises the private key of a chat participant can decrypt any previous chats ...


Mostly. The two problems are actually more closely equivalent in a gap model than in a non-gap model. Square-DH clearly reduces to CDH either way, but CDH reduces to two calls to Square-DH (you have 3, but you can use $(u-v)^2$ to make it 2). This is fine if the Square-DH adversary is always right, but maybe the adversary only solves the Square-DH problem ...

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