[Back] With pair-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:

\(e(U_1+U_2,V_1) =e(U_1,V_1) \times e(U_2,V_1)\)

\(e(U_1,V_1+V_2) =e(U_1,V_1) \times e(U_1,V_2)\)