How Powerful are the DDH Hard Groups?

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The question whether Identity-Based Encryption (IBE) can be based on the Decisional Diffie-Hellman (DDH) assumption is one of the most prominent questions in Cryptography related to DDH. The authors study limitations on the use of the DDH assumption in cryptographic constructions, and show that it is impossible to construct a secure Identity-Based Encryption system using, in a black box way, only the DDH (or similar) assumption about a group. Their impossibility result is set in the generic groups model, where they describe an attack on any IBE construction that relies on oracle access to the group operation of randomly labeled group elements - a model that formalizes naturally DDH hardness.