With pairing-based cryptography we have two cyclic groups (\(\mathbb{G_1}\) and \(\mathbb{G_2}\)), and which are of an order of a prime number (\(n\)). A pairing on \((\mathbb{G_1},\mathbb{G_2},\mathbb{G_T})\) defines the function \(e:\mathbb{G_1} \times \mathbb{G_2} \rightarrow \mathbb{G_T}\). If \(U\) is a point on \(\mathbb{G_1}\), and \(V\) is a point on \(\mathbb{G_2}\), we get [article]:
\(\hat{e}(aU,V) = \hat{e}(U,aV) = \hat{e}(U,V)^a\)
In this case, Bob will send Alice a signencryption message using her identity, and she will be able to decrypt it.