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Yeah, the question is something like "A customer wants to to secure their communications with <company> but the customer can only use a DES crypto system and does not have a shared DES key with <company>. How can they securely communicate without help from a third party?" Well, the customer could call the company and arrange for a meeting ...


The fact that $g$ is a generator (or not) of the group of inverse elements $G={\bf F}_p^{*}$, indeed does not affect the relation you wrote. But, if you want to apply Diffie-Hellman in a secure way, the order of $g$ has to be large. Say you choose a large prime $p$ (at least $1024$ bits). If $g$ is not a generator of $G$ then the order of $g$ shall divide ...


Fkraiem's answer is correct: this is not necessary. From your comment on his answer it seems however you don't understand why Alice and Bob retrieve the same key. This, again, doesn't rely on $g$ being a generator. Recall from your high school math classes that $(g^a)^b = g^{ab} = g^{ba} = (g^b)^a$. This is basically the trick that is being used here. Since ...


It does not; the equation holds for any element $g$. The fact that $g$ is a generator means only that every element of the group can be obtained a key. This is not at all necessary for the protocol.

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