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The authors develop an abstract framework that encompasses the key properties of bilinear groups of composite order that are required to construct secure pairing-based cryptosystems, and they show how to use prime-order elliptic curve groups to construct bilinear groups with the same properties. In particular, they define a generalized version of the subgroup decision problem and give explicit constructions of bilinear groups in which the generalized subgroup decision assumption follows from the decision Diffie-Hellman assumption, the decision linear assumption, and/or related assumptions in prime-order groups.
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