\(e(U_1+U_2,V_1) =e(U_1,V_1) \times e(U_2,V_1)\)
\(e(U_1,V_1+V_2) =e(U_1,V_1) \times e(U_1,V_2)\)
If \(U\) is a point on \(G_1\), and \(V\) is a point on \(G_2\), we get:
\(e(aU,bV) = e(U,V)^{ab}\)
If \(G_1\) and \(G_2\) are the same group, we get a symmetric grouping (\(G_1 = G_2 = G\)), and the following commutative property will apply:
\(e(U^a,V^b) = e(U^b,V^a) = e(U,V)^{ab} = = e(V,U)^{ab}\)
In this example, we will prove these mappings for a pairing.