Pairings

Let $G_1$, $G_2$, and $G_t$ be three cyclic groups of the same order. A map $e : G_1×G_2 → G_t$ is said to be bilinear if $\forall u, v ∈ G;\forall a, b ∈ \{0, . . . , |G|−1\}: e(u^a, v^b) = e(u, v)^{ab}$. If a bilinear map $e$ is also non-degenerate (meaning, it does not map all pairs in $G_1 × G_2$ to the identity element $1_{G_t}$) and $e$ is efficiently computable, then $e$ is called a pairing. This terminology refers to the fact that $e$ associates each pair of elements in $G_1 × G_2$ to an element of $G_t$ .

In the general case that $G_1 \neq G_2$, the pairing is said to be asymmetric, while the case that $G_1 = G_2$ is called symmetric. Asymmetric pairings are much more efficient in practice than symmetric pairings.