Diffie-Hellman on additive group Given the finite cyclic, additive group (G, +), with |G| = n and generator = g, what are the computations and exchanged messages for Diffie-Hellman?

What I tried myself:


Alice chooses a private $a$ and sends $p(|G|)$ and $g$ (generator) to Bob.
Alice calculates $A = a\cdot g  \mod p (|G|)$ and sends it to Bob.
Bob chooses a private $b$ and calculates $B = b \cdot g \mod p (|G|)$ and sends it back to Alice.
Alice calculates $a \cdot B \mod p (|G|)$ which is the shared key.
Alice calculates $b \cdot A \mod p (|G|)$ which is the shared key.


Is this way of thinking correct? Because I'm not sure because the group is additive.
Given the additive group $(G, +)$ with $|G| = p$ and generator $P$, what are the computations and exchanged messages for Diffie-Hellman?

I use order $p$ and assuming $p$ to be prime and the generator as $P$ (as it is used in context of elliptic curve groups - since you need additive groups where the DLP and the CDHP are hard - which is not the case for $\mathbb{Z}_n$).


Alice chooses random $a\in \mathbb{Z}_p^*$ and sends $A=aP$ (this means $P+\ldots+P$, $a$ times).
Bob chooses  $b\in \mathbb{Z}_p^*$ and sends $B=bP$.
Alice computes $K=aB=(ab)P$
Bob computes $K=bA=(ba)P=(ab)P$


Note that since I assume that we work on elliptic curve groups there is no reduction modulo the group order after the additive operation.