I seeing a paper about Elliptic curve based proxy re-encryption.
And I want to implement this through BLS12-381 Curve. However, When looking at the documentation for paring or the library, the value of $F_{q^{12}}$ is output as the output of pairing.
The paper requires:
Let e : $G_1$ × $G_1$ → $G_2$ be a bilinear map, z = e($G_1$, $G_1$) ∈ $G_2$
And need to compute (zrG + Pm)
How can I multiply z in $F_{q^{12}}$ and $rG$ Point in $F_{q}$from "$z$ $rG$"?
Should I replace $F_{q^{12}}$ with $F_{q}$? If so, how?
And please let me know what to look for to get relevant knowledge.