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We ask whether an Identity Based Encryption (IBE) system can be built from simpler public-key primitives. We show that there is no black-box construction of IBE from Trapdoor Permutations (TDP) or even from Chosen Ciphertext Secure Public Key Encryption (CCA-PKE). These black-box separation results are based on an essential property of IBE, namely that an IBE system is able to compress exponentially many public-keys into a short public parameters string.