Diffie Hellman groups I saw that non-negative integers with the addition operation cannot be the Diffie Hellman group. I'm having trouble understanding why it cannot be the DHKE group.
To be DHKE group, there are five properties to be held,

The operator is closed
associativity
identity element exists
inverse element exists
commutativity

I tried with the group of {0,1,2,3,4,5,6} and p=7, and all the above properties were held. I couldn't find any appropriate example to show the above properties are not held. What other examples can show that the non-negative integers with addition operator is not one of Diffie Hellman key 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.