With pairing-based cryptography we have two cyclic groups (\(G_1\) and \(G_2\)), and which are of an order of a prime number (\(n\)). A pairing on \((G_1,G_2,G_T)\) defines the function \(e:G_1 \times G_2 \rightarrow G_T\), and where \(g_1\) is a generator for \(G_1\) and \(g_2\) is a generator for \(G_2\). If we have the points of \(U_1\) and \(U_2\) on \(G_1\) and \(V_1\) and \(V_2\) on \(G_2\), we get the bilinear mapping of:
\(\hat{e}(U_1+U_2,V_1) =\hat{e}(U_1,V_1) \times \hat{e}(U_2,V_1)\)
\(\hat{e}(U_1,V_1+V_2) =\hat{e}(U_1,V_1) \times \hat{e}(U_1,V_2)\)
If \(U\) is a point on \(G_1\), and \(V\) is a point on \(G_2\), we get:
\(\hat{e}(aU,bV) = \hat{e}(aU,bV) = \hat{e}(U,V)^{ab}\)
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:
\(\hat{e}(U^a,V^b) = \hat{e}(U^b,V^a) = \hat{e}(U,V)^{ab} = \hat{e}(V,U)^{ab}\)
In this example, we will prove these mappings for a pairing.