Suppose that there are $n+1$ parties - $B,A_1,A_2,...,A_n$ that want to share a secret key

The protocol of exchanging is roughly the same as Diffe-Hellman

Chose a group $G$ with an order of $p$ - a prime number and a generator element $g$

• Each $A_i$ generates a random number $a_i \in \{1,...,q\}$ and send $B$ the value $X_i \leftarrow g^{a_i} $
• B generates a random number $b \in \{1,...,q\}$ and send each $A_i$ the value $Y_i \leftarrow X_i^b$

The shared key of the group is $g^b$

It is obvious that $B$ can calculate this key.

How can each $A_i$ calculate the shared key ?

I tried finding the inverse $a_i^{-1}$ of $a_i$ in mod $q$ and then taking both sides of $Y_i=g^{a_ib}$ to the power of $a_i^{-1}$ but when I plug in real numbers, it does not seem right.