KP-ABE Key Generation

In Key Policy Attribute-based encryption if we need a targeted broadcast where the set of attributes are the same for a group of users. It means they all share the same secret key right? Or is it that every user needs to have at least 1 attribute value different from all other users.

Example: If I need to encrypt a message to everyone who is a researcher in CS then attribute= researcher dept=CS. Then in this case all share the same secret key.

I read your previous post also. In KP-ABE, the key space $K$ the set of all poly-sized boolean formulas $φ(\vec z)=φ(z_1, z_2, ..., z_n)$, so $φ$ is essentially a boolean function that is encoded into each user's key.

In a CS department for example, each person's key may have the following boolean formula format encoded into its key $φ(\text{name, surname, rank, department}) = name \lor surname \lor rank \lor CS$.

In KP-ABE each message can be decrypted iff the predicate $P_n$ = 1, where P_n is defined on $K$ x $I$ as

$P_{n}(\phi \in K \backslash\{\epsilon\}, \text { ind }=\vec{z} \in I):= \begin{cases}1 & \text { if } \phi(\vec{z})=1, \text { and } \\ 0 & \text { otherwise }\end{cases}$

and each plaintext is a pair $(ind = \vec z, m)$

You can read more on the subject on this [BSW10]