Diffie-Hellman key exchange using addition instead of multiplication?
Why can't we use $(Z_p,+)$?

In the DH key Exchange Alice and Bob agree on a common generator g which allow to share a common key. In fact in $(Z_p,+)$ every non zero element is a generator.

If a common generator g is preallably chosen, when Alice chose a secret a, transmitting A=g.a, the secret a will be immediatly disclosed by any one in the public channel by simply Computing $a=\frac{A}{g}$ and key exchange will be violated which isn't the case in $(Z_p,*)$, as the inversion by the discret logarithm, is considered hard.