How is it decided if $G_1$ and $G_2$ are two “additive” or “multiplicative” cyclic groups? According to wiki's definition of Bilinear pairing…


  Let $G_1$ and $G_2$ be two additive cyclic groups of prime order $q$, and $G_T$ another cyclic group of order $q$ written multiplicatively. A pairing is a map: $e: G_1 \times G_2 \rightarrow G_T$
  but, there also have different definition such that $G_1$ and $G_2$ be two multiplicative cyclic groups.


How is the decision made if $G_1$ and $G_2$ are two additive cyclic groups or multiplicative cyclic groups?
Notation is basically a free choice of the author, as they describe functionally the same. And there is no fixed definition for this. However, common practice in mathematical publications is:


Multiplicative notation for arbitrary groups
Additive notation for commutative groups


This can be found here: math-SE,  wolfram

Wikipedia also states, that additive notation is primarily used for modules and rings. However, that is quite obvious because for those you need both types of notations.



Interestingly, in cryptography we rarely see any non-abelian groups except matrix multiplication, yet multiplicative notation is quite common. The emphasis here might be on the difficulty of the discrete logarithm problem: If a group has the property, that calculating DLOG is difficult, we tend to use multiplicative notation. Additionally, we note generic groups in multiplicative notation mostly.

Additive notation is used mostly just for elliptic curves, or of it is actually a group operation derived from the addition of integers. In that case it might be used to emphasize: In this group the DLOG problem is easy.