Algorithm 1 Energy Efficiency Optimization Algorithm for Secure Multicast Transmission

Input: The beam domain channel transmit statistics CSI ${\mathsf{\Omega }}_k$ and ${\mathsf{\Omega }}_{\mathrm{ev}}$, initial power allocation ${\mathsf{\Lambda }}^{\left(0\right)}$, the threshold $ϵ$

Output: Optimal power allocation matrix $\mathsf{\Lambda }$

1: Initialization: $\ell =0$

2: Calculate ${\mathsf{\Lambda }}^{\left(0\right)}$ using (26), (27), and (33)

3: repeat

4:    Update $\ell \leftarrow \ell +1$

5:    Initialization: $t=0$, ${\mathsf{\Gamma }}^{\left(0\right)}={\mathsf{\Lambda }}^{(\ell -1)}$, ${\eta }^{\left(0\right)}$ using (31)

6:    repeat

7:     Update $t\leftarrow t+1$

8:     Calculate ${\mathsf{\Gamma }}^{\left(t\right)}$ by using (30) with ${\eta }^{\left(t-1\right)}$

9:     Update ${\eta }^{\left(t\right)}$ using (31)

10:    until $\parallel {\mathsf{\Gamma }}^{\left(t\right)}-{\mathsf{\Gamma }}^{(t-1)}\parallel \le ϵ$

11:    Return ${\mathsf{\Lambda }}^{\left(\ell \right)}={\mathsf{\Gamma }}^{\left(t\right)}$

12: until $\parallel {\mathsf{\Lambda }}^{(\ell )}-{\mathsf{\Lambda }}^{(\ell -1)}\parallel \le ϵ$

13: Return $\mathsf{\Lambda }={\mathsf{\Lambda }}^{\left(\ell \right)}$