I'm a bit confused regarding the bilinear pairing operation. Let's say I have a Public key of a receiver $P_r = P^x$ and I want to create a symmetric key using KEM with a pairing operation. If I chose $R = rP$ and compute $V = (Pr, P)^r $which results into $V = (P^x, P)^r = (P^x, rP) = (P^x, R)$. Here, I am confused about how can I solve $(P^x)$?
So, basically, my question is If $e(aP, bP) = e(P, P)^{ab}$ (an additive elliptic group to a multiplicative elliptic group) then how can we solve $e(P^a, P^b)$? Moreover, $e(P^a, P^b)$ will be an additive group or multiplicative group?
