Identity-Based Encryption Protocols Using Bilinear Pairing

model without random oracles. Additionally, we show that CCA security can also be achieved
through the reduction to oracle decision bilinear Diffie-Hellman problem (OBDH).
We note that Naccache [77] has independently obtained a similar construction as of ours
in Section 5.2. Though the construction is similar, the work by Naccache does not perform
any concrete security analysis. In fact, the Naccache work asserts that the loss of security
due to the generalisation is “insignificant”. As discussed in Section 5.3, this is not correct. In
fact, the conversion of security degradation into a trade-off between time and space is original
to our work and is the most important feature of the generalisation of Waters scheme. On
the other hand, we would like to mention that Naccache’s work presents a clearer probability
analysis than that of Waters.
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