# Equality Check with EC Point Division using Bilinear Pairing

I'm currently trying to implement a PEKS scheme for my master's thesis and got stuck on a check I have no clue how to implement.

The equation looks like this: $$\hat{e}\left(P_1, T_3\right)\stackrel{?}{=}\frac{\hat{e}\left(T_1,T_2\right)}{\hat{e}\left(T_1,T_3\right)}$$ Note: $$\hat{e}$$ is a bilinear pairing function $$G_1\times G_2\rightarrow G_t$$ and formulas are using the notation $$P_1=g_1^s$$.

My problem is that I don't understand how to perform the division of two EC points without having access to any scalar of $$P_1, T_1, T_2, T_3$$. I guess I am missing something fundamental here. Does anyone know how to achieve such a check?

Regards, Michael

• For most commonly used pairing operations, the group $G_t$ is not an elliptic curve group; instead, it is a finite field extension group Aug 24 at 11:42
• Hi, thanks for your response. It looks like you are right for the BLS12-381 pairing I am using. The implementation of $G_t$ uses an extended finite field $F_{12}$. I implemented a division operator on $G_t$ as $(n, d)\mapsto n*d^{-1}$ where $d^{-1}$ is the multiplicative inverse, and all seems to work. If you want to push your comment as an answer, I would be glad to mark it as solving. Aug 24 at 18:03

For most commonly used pairing operations, the group $$G_t$$ is not an elliptic curve group; instead, it is a finite field extension group.
You later mention that you are using BLS-381; in that case, $$G_t$$ is $$GF(p^{12})$$ (for a specific 381 bit prime $$p$$); in that field, you can perform division as you stated...