Simple example for Bilinear mapping

Question

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$.

Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy two properties.

1) $e(aP,bQ) = e(P,Q)^{ab}$ for all $P,Q \in \mathbb{G} , a,b \in \mathbb{Z}_q$

2) $e(P,P) \ne 1_{\mathbb{G}_T}$ for all $P \in G$

In addition to the above two, it has to be computable.

Is there any simple example for such bilinear pairing?

I searched over internet as I can but I didn't find.

I want example like this $e(x,y) = xy \mod n$ (It's not a valid example).

Note: The sets under consideration $\mathbb{G},\mathbb{G}_T$ must have at least four elements.

Answer 1

Take $\mathbb{G}=\mathbb{Z}_q$; take $e(1,1)$ to be any non-trivial element of $\mathbb{G}_T$, and define $e(P,Q)=e(1,1)^{PQ}$.