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This scheme is insecure, as anyone with the public key can generate a forgery of an arbitrary message. To do this, the forger would take the message $M$, the public key $y$, pick an arbitrary $z$, and compute $r = y^{-H(M)} g^{z} \bmod p$ and output $(r,z)$


1

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 ...


1

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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