Choosing between additive or multiplicative group for Diffie-Hellman I am trying to construct a theoretical Diffie-Hellman key exchange protocol. However, i cannot understand the difference between choosing an additive group or a multiplicative group. I believe an additive would make it much more simple, but would a multiplicative group make it more difficult to break? What would be the difference between the use of those 2 kinds of number groups?
From an abstract algebraic point of view the two are one and the same, a group.

However, depending on how that group is defined additional structure may make additional operations possible or easy that are detrimental to your security.

For example:
In a finite field of prime order calculating logarithms is hard but division is easy.

More mathematically the two problems are


calculating $x$ given $g$ and $g^x$
calculating $x'$ given $g$ and $x' \cdot g$


and they have totally different complexities.

Both addition and multiplication in this case define a group but integer multiplication is not hard to invert.

And, the usual disclaimer, please don't design your own crypto unless for exercises. ;)