Given two IBE schemes:

Lets say, we have a message $m$ and encrypt it with both Boneh-Franklin IBE and Park-Lee IBE schemes, with the same $id$ (public keys of $id$ are calculated according to both key generation algorithms, using the same $msk=\alpha$ in $\mathbb{Z}_p$). The two IBE schemes use the same curve. Adversary is able to query two kinds of secret key oracles.

I believe the constructed scheme is secure, but it seems impossible to make a security reduction to the underlying IBE schemes. Since the challengers of two schemes would use different master secret keys, the simulator of our scheme can not answer secret key queries properly.

Are there any security implications for this scheme? How to prove its security?

  • $\begingroup$ The following idea may works. We can directly make a reduction to the DBDH problem, since a DBDH instance can also be regarded as a CBDH instance in the simulation. However, how to make a proof if there is no such a transformation between the two assumptions of underlying schemes? $\endgroup$
    – Ati
    May 8 '21 at 11:18

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