Generalizations of Verheul's theorem to asymmetric pairings
Koray Karabina, Edward Knapp, Alfred Menezes · Advances in Mathematics of Communications · 2013
For symmetric pairings $e : \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$, Verheul provedthat the existence of an efficiently-computable isomorphism $\phi : \mathbb{G}_T\rightarrow \mathbb{G}$ implies that the Diffie-Hellman problems in $\mathbb{G}$ and $\mathbb{G}_T$can be efficiently solved. In this paper, we explore the implications ofthe existence of efficiently-computable isomorphisms $\phi_1 : \mathbb{G}_T\rightarrow \mathbb{G}_1$ and $\phi_2 : \mathbb{G}_T \rightarrow \mathbb{G}_2$ for asymmetricpairings $e : \mathbb{G}_1 \times \mathbb{G}_2 \rightarrow \mathbb{G}_T$. We also give a simplifiedproof of Verheul's theorem.