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A cyclic group g of order p and an element h ∈ g . Task. Compute a ∈ ZZ p such that h = ga . We next define another well-known basic computationally hard problem for a cyclic group. Computational Diffie-Hellman (CDH). Instance. A cyclic group g of order p and a tuple (g, ga , gb ) where a and b are uniform random elements of ZZ p . Task. Compute gab . In other words, an algorithm (or an adversary) A for solving the CDH problem takes as input a tuple (g, ga , gb ) and has to output gab . The advantage of A in solving the CDH problem is defined as follows.

Just like in PKE, it is a formalisation of the adversary’s inability to distinguish between ciphertexts arising out of two equal length messages M0 and M1 . , the adversary’s goal is to compromise the security of the identity it chooses as the target identity. A random bit γ is chosen and the challenge ciphertext is produced by encrypting Mγ under the target identity. The adversary wins if it can predict γ with a probability significantly away from half. The main difference from PKE schemes is that a coalition of valid users of an IBE scheme can possibly launch an attack against another user of the scheme.

In the above sequence, the following points are to be noted 1. G0 is the game which defines the security of the protocol and so Adv(A ) = |Pr[γ = γ ′ ] − 1/2| = |Pr[X0 ] − 1/2|. 2. Gk is designed such that the bit γ is statistically hidden from the adversary. So, Pr[Xk ] = 1/2. 3. Games Gi−1 and Gi differ: a. the difference is not too much; b. the adversary should not be able to notice whether he is playing Game Gi−1 or Game Gi . 4. More precisely, Pr[Xi−1 ] − Pr[Xi ] is bounded above by a. either, the advantage of an adversary in breaking one of the smaller protocols; b.

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A density lemma

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