Pairings-based Cryptography (Part 1)

This post contains some basic facts about Pairing-based Cryptography. I write this post mainly for the reason to have a easy to find reference for myself and to recall some definitions. For readers that are more interested in pairings in context of cryptography, a good further reading source is the dissertation of Lynn [1], wherefrom i also adopted the usage of the multiplicative notation as a shortcut to represent $$\underbrace{a\circ a\circ ... \circ a}_{n\;times} = a^n$$ if $\circ$ is the group operation of $\mathbb{G}$ and $a \in \mathbb{G}$.

Definition [Pairing] Let $r$ be a prime number and $\mathbb{G}_1$ and $\mathbb{G}_T$ be cyclic groups of order $r$. Let $\mathbb{G}_2$ [not necessary cyclic] in which every element has order $r$. Then a pairing is the map \begin{equation} e: \mathbb{G}_1 \times \mathbb{G}_2 \rightarrow \mathbb{G}_T \end{equation} and which has the properties ($\mathsf{e}$ is the neutral element in the group): (Non-Degeneracy) $e(g_1,g_2) = \mathsf{e}_{\mathbb{G}_T}$ for all $g_2\in\mathbb{G}_2$ if and only if $g_1 = \mathsf{e}_{\mathbb{G}_1}$ (Non-Degeneracy) $e(g_1,g_2) = \mathsf{e}_{\mathbb{G}_T}$ for all $g_1\in\mathbb{G}_1$ if and only if $g_2 = \mathsf{e}_{\mathbb{G}_2}$ (Bilinearity) $e(g_1^a,g_2^b) = e(g_1,g_2)^{ab}$ for all $g_1\in\mathbb{G}_1$ and $g_2\in\mathbb{G}_2$ for all $a,b\in\mathbb{Z}$.