Figure 4: Games $H_0-H_3$ and $H_{4,0}-H_{4,L}$ for the proof of Lemma 3.8. $\mathbf{RF}_k := {0,1}^k \to \mathbb{Z}_p$ is a truly random function. $\mu_i|_k$ is the first $k$ bits of $\mu_i$.
Proof. The only difference between $H_{4,0}$ and $H_3$ is the simulation of $\text{crs}1$, which is generated by either BG (in $H_3$) or HG (in $H{4,0}$) since $\textbf{RF}_0(\epsilon) = x_0$ and $\mu_j|_0 = \epsilon$ for all $j \in [q_s]$. From that, we obtain a straightforward reduction to CRS indistinguishability of GS. ■
Lemma 3.17 ($H_{4,k}$ to $H_{4,k+1}$). There exist adversaries $\mathcal{B}_1$ against CRS indistinguishability of GS and $\mathcal{B}2$ against IND-mCPA security of PKE with running times $T(\mathcal{B}1) \approx T(\mathcal{B}2) \approx T(\mathcal{A})$ and Adv$H{4,k}$ – Adv$H{4,k+1} \le 4\text{Adv}{\text{GS}}^{\text{crsind}}(\mathcal{B}1) + 6\text{Adv}{\text{PKE}}^{\text{mcpa}}(\mathcal{B}_2) + \frac{2q_s}{p}$
Proof. We define the games between $H_{4,k}$ and $H_{4,k+1}$ in Figure 5 and an overview of the game transitions is presented in Figure 6.