Generalizations of verheul's theorem to asymmetric pairings

For symmetric pairings e: G × G → GT, Verheul proved that the existence of an efficiently-computable isomorphism Φ: GT → G implies that the Diffie-Hellman problems in G and GT can be efficiently solved. In this paper, we explore the implications of the existence of efficiently-computable isomorphisms Φ1: GT →G1 and Φ2: GT →G2 for asymmetric pairings e: G1 × G2 → GT. We also give a simplified proof of Verheul's theorem. © 2013 AIMS.