Algorithm 1 Encrypt messages by $U_A$.

Require: Shared PHI message $\mathbf{MES}$ and $(\mathbf{A},\rho ,\Gamma )$.

1: $U_A$ chooses a random vector $v=(s,r_2,···,r_n)\in {\mathbb{Z}}_p^n$.

2: $U_A$ also chooses $r_x\in {\mathbb{Z}}_p$, for $1\le x\le l$.

3: $U_A$ computes:

$C_1=\left\{\begin{array}{c}\widehat{C}=u^{H\left(\mathbf{MES}\right)};\hfill \\ \widetilde{C_1}=\mathbf{MES}·e{(g,g)}^{\alpha s};\hfill \\ C_1^{\prime }=g^s;\hfill \\ C_{1,x}=g^{a{M}_i·v}{\left(T_{\rho \left(x\right)}^{H\left(t_{\rho }\left(x\right)\right)}\right)}^{-r_x};\hfill \\ D_{1,x}=g^{r_x};\hfill \end{array}\right.$

4: $U_A$ computes $C_2=Si{g}_{s{k}_{U_A}^j}\left(C_1\right)$.

5: For each $I\subseteq \{1,···,l\}$, which satisfies ${\sum }_{i\in I}{w}_i{M}_i=(1,0,···,0)$, extract the corresponding attribute values ${\Gamma }_I=(t_{\rho \left(1\right)},···,t_{\rho \left(i\right)})$.

6: Compute $H\left({\Gamma }_I\right)$ for all $I\subseteq \{1,···,l\}$.

7: Construct the Bloom filter $BF$ by using $H\left({\Gamma }_I\right)$.

8: $U_A$ broadcasts message $C=<BF,(\mathbf{M},\rho ),C_1,C_2>$.