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: \(e(aU,V) =e(U,aV)\).
In this case we will convert:
\(e(X,a) \cdot {e(X,b)}^m \cdot {e(X,B)}^r =e(g_1,c)\)
into:
\(e(X,a) \cdot {e(X,b)}^m \cdot {e(X,B)}^r \cdot e(g_1,-c)=1\)