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Suppose the server did not include $v$ in the computation of $B$. In such case the following events have happened: The server has sent a salt value $s$ to the client. We might assume it is authentic (because it is easy for the fake server to get it from the real server). The client re-calculates its long term private key $x$ such that $v = g^x$. The client ...


But this proof value must be something both the client and the (legitimate) server can compute, and thus it must be entirely determined by: values chosen by the client and sent to the server during the authentication process, values chosen by the server and sent to the client during the authentication process, and the password and/or the ...

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