Diffie-Hellman insecure on addition modulo $n$ Assume that the group $G$ is the set $\mathbb{Z}_{n} = \{0,\ldots, n-1\}$ for a 1024 bit integer and $+$ is addition modulo $n$. Then why would Diffie-Hellman key exchange in this group be insecure?
With addition and $\mathbb{Z}_n$, each party chooses a secret $x$ and sends $xg \pmod n$ over the wire, for an agreed upon generator $g$. Division by $g$ modulo $n$ is easily computable, and reveals $x$.

In other words, a prerequisite for DH to be secure is that the equivalent to discrete logarithm is hard in the chosen group. With $\mathbb{Z}_n$ and addition, discrete logarithm becomes totally easy.