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I use multiplicative notation for $\mathbb{G}_1$ and $\mathbb{G_2}$ (as in your question) as this may make it a bit clearer for you. Lets say the groups are of prime order $p$ and and lets consider your elements w.r.t. bases $g_1$ and $g_2$. Lets write your elements $x_1$ and $x_3$ as $g_1^{a_1}$ and $g_1^{a_2}$ and $x_2$ and $x_4$ as $g_2^{b_1}$ and ...


3

For clarity, I will use an additive notation in the groups $\mathbb{G}_1$ and $\mathbb{G}_2$, and denote group elements with the "$x$" letter, not "$e$". For all $x_1, x_3 \in \mathbb{G}_1$ and $x_2, x_4 \in \mathbb{G}_2$, we have the following: \begin{eqnarray*} e(x_1 + x_3, x_2 + x_4) &=& e(x_1, x_2 + x_4) e(x_3, x_2 + x_4) \\ &=& e(x_1, ...



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