In identity-based broadcast encryption, suppose the broadcast ciphertext $(r_1,r_2,\dots,r_i,U,W,V)$ is sent to the receiver.

In some cases, they use some function $f(x)=k-\prod_{i=1}^{t} (x-v_{i})$ to get $k$ by replacing $v_{i}^{'}$ in place of $x$. In this case, only one function for $t$ receivers is possible.In real implementation, how to define such kind of function for unknown $x$ for $n$ receivers.
However, In decryption, $$k^{'}=f(v_{i}^{'})=\sum_{j=0}^{t-1}c_{j}(v_{i}^{'})^{j}+(v_{i}^{'})^{t} mod \space q $$ where $v_{i}^{'} $ is calculated using $H_{1}(e(V,d_{i}))$. That means $v_{i}^{'} $ value is different.
If so,the same key $k$ is impossible for all receivers.
Here, $t$ different functions for $t$ receivers?If so, $t$ different W values.

Example paper: Anonymous Multi-Receiver Identity-Based Authenticated Encryption with CCA Security by Fan et al.


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