D. Boneh, P. Papakonstantinou, C. Rackoff, Y. Vahlis and B. Waters, “On the Impossibility of Basing Identity Based Encryption on Trapdoor Permutations,” 2008 49th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 283-292, doi: 10.1109/FOCS.2008.67.
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.