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The authors construct two efficient Identity-Based Encryption (IBE) systems that admit selective-identity security reductions without random oracles in groups equipped with a bilinear map. Selective-identity secure IBE is a slightly weaker security model than the standard security model for IBE. In this model the adversary must commit ahead of time to the identity that it intends to attack, whereas in an adaptive-identity attack the adversary is allowed to choose this identity adaptively. The first system - BB1 - is based on the well studied decisional bilinear Diffie-Hellman assumption, and extends naturally to systems with hierarchical identities, or HIBE. The second system - BB2 - is based on a stronger assumption which they call the Bilinear Diffie-Hellman Inversion assumption and provides another approach to building IBE systems.
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