Mathematical Prove (3)

Bilinear Pairing over Composite Group

Given groups $\mathbb{G}$ and $\mathbb{G}_T$ of order $N=pq$ where $p,q$ are prime, and a bilinear pairing $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$.

The subgroups of $\mathbb{G}$ with size $p,q$ are defined as $\mathbb{G}_p$ and $\mathbb{G}_q$. Let generator of $\mathbb{G}$ be $g$. The generators of $\mathbb{G}_p$ and $\mathbb{G}_q$ be $g_p$ and $g_q$ can be computed as $g_p = g^q \in \mathbb{G}_p$ and $g_q = g^p \in \mathbb{G}_q$

In addition to basic bilinear pairing properties, bilinear pairing over composite group holds some additional properties:

• $e(g_p,g_q) = e(g^q,g^p) = e(g,g)^{pq} = e(g,g)^N = 1$
• $e({g_p}^a{g_q}^b,{g_p}^c) = e(g^{qa+pb}, g^{qc}) = e(g,g)^{q^2ac+Nbc} = e({g_p}^a,{g_p}^c)$