Diffie Hellman groups
To be DHKE group, there are five properties to be held,

This is true; however, to be a secure DHKE group, there has to be an additional property:

The &quot;discrete log problem&quot; needs to be hard; that is, given a public value $xG$ (where $G$ is the public group identifier, $x$ is your private value, and $xG$ is the generator acted upon itself $x$ times), it is hard to recover $x$.

In the addition case, we have $xG = \underbrace{G + G + … + G}_{x \text{ times}} = x \times G$., where $\times$ is integer multiplication.
If we know $x \times G$ and we know $G$, then it is easy to recover $x$ by doing a simple division (modular division if we're doing the addition modulo $p$, which isn't much harder), and hence the Diffie-Hellman would be trivially insecure.
Actually, just having a secure discrete log problem is not sufficient; the &quot;Diffie-Hellman&quot; problem also needs to be hard, that is, given $G$, $xG$ and $yG$, it should be hard to recover $xyG$ (this is known as the computational Diffie Hellman problem), or given $G$, $xG$, $yG$, $zG$, it should be hard to determine whether $xy = z$ (this is known as the decisional Diffie Hellman problem).  Either of these problems may be easier than the &quot;discrete log problem&quot; for any group; however, for addition, none of these problems are hard.