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How come that in various articles about pairings I never saw anybody mention that there is a possibility to turn Type 2 pairing into Type 1 by setting $e' : G_2 \times G_2 \rightarrow \mu, (P,Q) \mapsto e(Tr(P), Q)$, where $e : G_1 \times G_2 \rightarrow \mu$ is Type 2 pairing and $Tr$ is the trace map?

The only drawbacks I can see is that points from $G_2$ have rather nasty representation and there is performance loss associated with the need to calculate $Tr$. Is it enough to not consider such object or are there other problems with this approach?

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I think there is some complexity problems with hashing into $G_2$ in type-2 pairings. However, That should not be as oblvious as it is needed for not mentioning this in literature. Maybe there are other reasons. – Habib Nov 6 '14 at 18:01

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