@ArtjomB. I ask here because I think some other people will know about something that I don't know or something I have wrongly misunderstood. Main point is why they do not consider and not why they publish. Just querying my thinking.

Artjom B. In the above problem, if $t=a$ then, the attacker achieves in decryption oracle. As $s$ value is given, the problem lies in finding $a$ value. Can you please explain suitable assumption for finding value of $e(d_{A},ID_{B})$ , given $ID_{A},ID_{B}$??

Artjom B. Can you please explain suitable assumption for finding value of $e(d_{A},ID_{B})$ , given $ID_{A},ID_{B}$??

$$\begin{eqnarray} & = & 1 - (\frac{q}{q} - \frac{1}{q}) \\ & = & \frac{1}{q} \approx 0. \end{eqnarray}$$ As I calculated the parenthesis first. $\frac{q-1}{q}$ That is why I got (1-0.9=0.1)

Let us continue this discussion in chat.

Artjom B. why $2^{170}!$ ? In the formula described above, just $r$! I wanna know substituted value in p(n) formula.

Artjom B. Can you please explain calculating the probability? I can't get that answer. $3.10^{-42}$. Suppose server 2 wants to try to get the same master secret key with server 1 without notice of server 1. First, it will get private key corresponding to ID1 from server 1. And then, it may try repeatedly to get the same private key using ID1. Is it easy or not?

fgrieu. What about the case of IBE servers that are not well-behaving? In such cases, security risk?

ArtjomB. Can that probability cause weak security?