# Probability of generating same master secret key in Identity-based Encryption

Suppose multiple servers use same IBE domain parameters (I mean same curve description parameters and field) for master secret key setup. Is there any possibility for generating the same system parameters?

Take for example the BasicIdent IBE scheme, by Boneh and Franklin. Let us assume that the curve, pairing and groups are fixed. During the Setup step, one has to select a random generator $P \in \mathbb G_1$ and a random master key $s \in \mathbb Z_q^*$. We can assume that these parameters are sampled uniformly.
The probability of selecting the same parameters in a different domain mostly depends on selecting the same $P$ and $s$. The order of both $\mathbb G_1$ and $Z_q^*$ is $q$, so the probability of generating the same system parameters is $\frac{1}{q} \cdot \frac{1}{q} = \frac{1}{q^2}$, which is negligible.