Compared to $S_{0,k,6'}$, there are more valid forgeries in $S_{0,k,7}$ and we have
Thus, $\mathrm{Adv}{S{0,k,6}} - \mathrm{Adv}{S{0,k,7}} \leq \frac{q_s}{p}$ and we conclude the lemma. ■
Lemma 3.39 (S${0}$${,k,}$${7}$ to S${0}$${,k,}$${8}$). $\mathrm{Adv}{S{0,k,8}} = 2\mathrm{Adv}{S{0,k,7}}.$
Lemma 3.40 (S${0}$${,k,}$${8}$ to S${0}$${,k,}$${9}$). There exists an adversary $\mathcal{B}$ against IND-mCPA security of PKE with $T(\mathcal{A}) \approx T(\mathcal{B})$ and $\mathrm{Adv}{\mathrm{PKE}}^{\mathrm{mcpa}}(\mathcal{B}) \geq |\mathrm{Adv}{S_{0,k,9}} - \mathrm{Adv}{S{0,k,8}}|$.
Lemma 3.41 (S${0}$${,k,}$${9}$ to S${0}$${,k,}$${10}$). $\mathrm{Adv}{S{0,k,10}} = \mathrm{Adv}{S{0,k,9}}.$
Lemma 3.42 (S${0}$${,k,}$${10}$ to S${0}$${,k}$−${1}$). $\mathrm{Adv}{S{0,k-1}} = \mathrm{Adv}{S{0,k,10}}.$
Summarizing the above lemmata, we have $\mathrm{Adv}{S{0,k}} - \mathrm{Adv}{S{0,k-1}} \leq 4\mathrm{Adv}_{\mathrm{GS}}^{\mathrm{crsind}}(\mathcal{B}1) + 6\mathrm{Adv}{\mathrm{PKE}}^{\mathrm{mcpa}}(\mathcal{B}_2) + 2\frac{q_s}{p}$ and conclude Lemma 3.31. ■
By defining $\mathbf{RF}_0(\epsilon) := x_0 \stackrel{$}{\leftarrow} \mathbb{Z}_p$, similar to Lemma 3.16, we have
Lemma 3.43 (S${0,}$${0}$ to S$_1$). There exists an adversary $\mathcal{B}$ against CRS indistinguishability with running time $T(\mathcal{A}) \approx T(\mathcal{B})$ and $\mathrm{Adv}{\mathrm{GS}}^{\mathrm{crsind}}(\mathcal{B}) \geq |\mathrm{Adv}{S_1} - \mathrm{Adv}{S{0,0}}|$.
Similar to Lemmata 3.14 and 3.15, we have
Lemma 3.44 (S$_1$ to S$_2$). $\mathrm{Adv}{S_2} = \mathrm{Adv}{S_1}.$
Lemma 3.45 (S$_2$ to S$_3$). There exists an adversary $\mathcal{B}$ against CRS indistinguishability with running times $T(\mathcal{A}) \approx T(\mathcal{B})$ and $\mathrm{Adv}{\mathrm{GS}}^{\mathrm{crsind}}(\mathcal{B}) \geq |\mathrm{Adv}{S_3} - \mathrm{Adv}_{S_2}|$.
Observing that $\mathbf{crs}0 \stackrel{$}{\leftarrow} \mathbf{BG(par)}$, $z{0,i} = z_{1,i} = x_0 + m_i$ and $\rho_0 \stackrel{$}{\leftarrow} \mathbf{P(\mathbf{crs}_0, ins_0, w_0)}$, we have $G_3 = S_3$ and
Lemma 3.46 (S$_3$ to G$_3$). $\mathrm{Adv}{G_3} = \mathrm{Adv}{S_3}.$
Summarizing Lemmata 3.30, 3.31 and 3.43 to 3.46, we have $\mathrm{Adv}{G_2} \leq \mathrm{Adv}{G_3} + (4L+3)\mathrm{Adv}_{\mathrm{GS}}^{\mathrm{crsind}}(\mathcal{B}1) + 6L \cdot \mathrm{Adv}{\mathrm{PKE}}^{\mathrm{mcpa}}(\mathcal{B}_2) + \frac{2Lq_s}{p}$.
We omit high level outlines of the game transitions in Lemmata 3.10 and 3.31 since they are very similar to Figures 3 and 6 for Lemmata 3.8 and 3.17.
4 Instantiation
We instantiate our generic construction in Type-III bilinear groups under the SXDH assumption. Through-out this section, we denote group elements in $\mathbb{G}_1$ with plain upper-case letters, such as $X$, and elements in $\mathbb{G}_2$ such letters with tilde, such as $\tilde{x}$. Scalar values in $\mathbb{Z}_p$ are denoted with lower-case letters. We may also put a tilde to scalar values or other objects when they are related to group elements in $\mathbb{G}_2$ in a way that is clear from the context.
We begin with optimizations in Section 4.1 made on top of the generic construction. We then present a concrete scheme for signing unilateral messages in Section 4.2 and for bilateral messages in Section 4.3 followed by full details of the Groth-Sahai proofs in Section 4.4.
4.1 ElGamal Encryption with Common Randomness
Observe that relation $(z_0 - z_1)(x_2 - z_2) = 0$ in $\mathcal{L}_1$ is a quadratic equation and it can be proved efficiently by a GS proof if $z_0$ and $z_1$ are committed in the same group and $z_2$ is committed in the other group. Relevant encryptions should follow the deployment of groups. We thus build the first two ciphertexts, $ct_0$ and $ct_1$ in $\mathbb{G}_1$, and $ct_2$ in $\mathbb{G}_2$.
To gain efficiency, we consider using the same randomness for making $ct_0$ and $ct_1$. For this to be done without spoiling the security proof, it is sufficient that one of the ciphertext $ct_b$ is perfectly simulated given the other ciphertext $ct_{b-1}$. Formally, we assume that there exists a function, say SimEnc, such that, for