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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.


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 ...


In cases where Alice and Bob are guaranteed to arrive at the same key, this is impossible: the function that takes Alice and Bob's private info as input, and produces the public transcript as output, must be a one-way function if the scheme is to be secure and if it always negotiates a shared key. If it sometimes fails, then you don't necessarily get a OWF; ...

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