Identity-Based Encryption Protocols Using Bilinear Pairing

plays the security game (h, n ′ )-M 1 with an adversary for (h, n)-H 1 . The adversary executes
the commitment stage; then the simulator sets up the HIBE based on the adversary’s commitment
as well as the DBDH instance. The simulator gives the public parameters to the
adversary and continues the game by answering all queries made by the adversary. In the
process, it randomly chooses a bit γ and encrypts M γ using the DBDH instance provided
as input. Finally, the adversary outputs γ ′ . Based on the value of γ and γ ′ , the simulator
decides whether the instance it received is real or random. Intuitively, if the adversary has
an advantage in breaking the HIBE protocol, the simulator also has an advantage in distinguishing
between real and random instances. This leads to an upper bound on the advantage
of the adversary in terms of the advantage of the simulator in solving DBDH.
We want to prove (h, n)-H 1 secure in model (h, n ′ )-M 1 , where 1 ≤ n ′ ≤ n. This means
that the public parameters of the HIBE depend on n, while the adversary commits to a set
I ∗ of size n ′ in the commit phase.
DBDH Instance: The simulator receives an instance (P, P 1 = aP, P 2 = bP, Q = cP, Z)
of DBDH.
The simulator now starts the security game for model M 1 . This consists of several
stages which we describe below. We will consider security against chosen plaintext attacks
and hence the adversary will only have access to the key extraction oracle O k .
Adversary’s Commitment: The adversary commits to a set I ∗ of size n ′ . The elements
of I ∗ are from Z p . We write I ∗ = {v1, ∗ . . . , vn ∗ ′}.
Set-Up:
Define a polynomial F (x) in Z p [x] as follows.
F (x) = (x − v1) ∗ · · · (x − vn ∗ ′) (7.5.1)
= x n′ + a n ′ −1x n′ −1 + · · · + a 1 x + a 0 (7.5.2)
where the coefficients a i ’s are in Z p and are obtained from the values {v1, ∗ . . . , vn ∗ ′}. Since
F (x) is a polynomial of degree n ′ over Z p and v1, ∗ . . . , vn ∗ are its n distinct roots, we have
′
F (y) ≠ 0 for any y ∈ Z p \ {v1, ∗ . . . , vn ∗ ′}. The coefficients of F (x) depend on the adversary’s
input and one cannot assume any distribution on these values. Define a n ′ = 1 and a n =
a n−1 = · · · = a n ′ +1 = 0.
For 1 ≤ i ≤ h, define another set of polynomials J i (x) each of degree n in the following
manner. Randomly choose b 0,1 , . . . , b 0,h , b 1 , . . . , b n from Z p . Define
J i (x) = b n x n + b n−1 x n−1 + · · · + b 1 x + b 0,i (7.5.3)
The public parameters P 3,i s and Q j s are defined in the following manner.
• For 1 ≤ i ≤ h, define P 3,i = a 0 P 2 + b 0,i P .
• For 1 ≤ j ≤ n, define Q j = a j P 2 + b j P .
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