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5

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.


5

Update: My previous answer, although technically true, didn't answer your question. The issue is that the strong-DDH is not hard when using pairing groups, so my answer was merely stating that your problem is at least as hard as an easy problem (duh!) After some thought, I realized your problem is at least as hard as the 2-weak Bilinear Diffie-Hellman ...


2

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 ...


1

Since $g$, $A=g^a$ and $B=g^b$ are made public, everyone could repeat the same operation and brute force DH with small exponents. There are already many responses: link to security.stackexchange.com link to crypto.stackexchange.com Summary: considering a security level of $n$, it is advised to use exponents of at least $2n$ bits.


1

DLP and factorization are very different problems (which cryptocipher gurus consider of same complexity). You can't really compare the choice of using a safe prime p in order to prevent the factorization of n=p*q (recommended for RSA) with the choice of using a prime p where (p-1)/2 has a large factor (recommended for DSA). Since you are interested with DH ...



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