Pairing map.

Let \(G_1\), \(G_2\) and \(G_3\) be three commutative groups. Then a pairing map is a function

This function takes pairs \((a_1,a_2)\) of elements from \(G_1\) and \(G_2\), and maps them to elements from \(G_3\) such that the bilinearity property holds, which means that for all \(a_1, b_1 \in G_1\) and \(a_2, b_2 \in G_2\) the following two identities are satisfied:

1. \(e(a_1\space.\space b_1, a_2) = \overbrace{e(a_1,a_2) \space .\space e(b_1,b_2)}^{\text{operator on}\space G_3}\)
2. \(e(a_1,a_2\space.\space b_2) = \underbrace{e(a_1,a_2) \space .\space e(a_1,b_2)}_{\text{operator on}\space G_3}\)

• Note that \(a_1\space.\space b_1\) is \(G_1\) group operator and \(a_2\space.\space b_2\) is \(G_2\) group operator.