mode crs$_0$ in the security proof (see Section 3.3 for details). This is achievable if $\rho_0$ is generated as a proof of “pairing product equations (PPEs)” (in both the real and simulated schemes). If the simulator has exponents, then $\rho_0$ is generated as a proof of “(linear) multiscalar multiplication equations”, which is more efficient than that of PPEs. We not only upgrade UF-XCMA to UF-CMA but also achieve an SPS scheme for vector messages by combining our xSPS with (partial) one-time signature at very low cost [3]. Thus, we select the UF-XCMA-secure scheme. See also Section 4 for efficiency.
3.2 Overview of Security Proof
Our main goal is to implement an additional check of $\mathcal{A}$'s forgery $\sigma^* := (\text{ct}_0^*, \text{ct}_1^*, \text{ct}2^*, \rho_0^*, \rho_1^*)$. We not only verify Groth-Sahai proofs, but also check $Z_0^* \in {\mathcal{G}^{x_0} \cdot M_i^{x_1}}{i=1}^{q_s}$ for $Z_0^* \leftarrow \text{Dec}(sk_0, \text{ct}_0^*)$. That is, we will force $\mathcal{A}$ to reuse an $M_i$ in queried messages for $Z_0^*$ (we will set $x_1 := 1$ to achieve this during the game transitions). With crs$_0$ for $\rho_0^*$ being in the perfect soundness mode, $\mathcal{A}$ is forced to fulfill $G^{z_0} = G^{x_0} \cdot M^*$. This leads to a contradiction and $\mathcal{A}$ never wins.
To change the success forgery condition, we replace the value $z_0 := x_0$ in signatures of the signing oracle and the additional forgery check with a value $z_0 := \mathbf{RF}_k(\mu|_k)$ where $\mathbf{RF}_k : {0,1}^k \to \mathbb{Z}_p$ is truly random, and $\mu|_k$ is the k-bit prefix of a random binary encoding $\mu \in {0,1}^L$ of a signed message $M \in \mathbb{G}$, where $L$ is the smallest even integer that is equal to or larger than the bit size of $p$. Note that encoding $\mu$ appears only in the security proof (not in the real scheme). We start with $\mathbf{RF}_0(\epsilon) := x_0$ for the empty string $\epsilon$. We will introduce more dependencies of $z_0$ on $x_2$ and $z_2^*$ in $\text{ct}_2^*$.
To increase the entropy of $z_0$ (this will make $z_0$ unpredictable for $M^*$ and force $\mathcal{A}$ to reuse $z_0$ from the signing oracle) and eventually set $z_0 := \mathbf{RF}_L(\mu)$, we replace $z_0 := \mathbf{RF}k(\mu|k)$ with $z_0 := \mathbf{RF}{k+1}(\mu|{k+1})$ step by step. At each step, we partition the signature space into two halves according to the $(k+1)$-th bit of $\mu$. The partitioning bit is dynamically changed by $z_2^*$ hidden in $\text{ct}_2^*$. At the beginning of the game, the simulator guesses the bit $z_2^*$ used in a forgery and aborts if the guess is incorrect ($z_2^*$ is accessible with the decryption key $sk_2$). Signature queries are created with a case distinction depending on the $(k+1)$-th bit $\mu[k+1]$ of $\mu$. If $\mu[k+1]$ is equal to the guessed $z_2^*$ from the forgery, nothing is changed in the signing process. However, if $\mu[k+1]$ is different from $z_2^*$, we use another independent random function $\mathbf{RF}'_k$ and set $z_1 := \mathbf{RF}'_k(\mu|_k)$ in the generated signature (i.e., more randomness is supplied).
Note that at this point, we want to change the encrypted values $z_0, z_1$ in the generated signature, while being able to decrypt the value $z_0^*$ from the forgery (to implement the additional check mentioned above). Intuitively, we can do so since the proved statement $(z_0 - z_1)(x_2 - z_2) = 0$ guarantees a consistent double encryption with $z_0 = z_1$ precisely when $x_2 \neq z_2$. Hence, if we initially set up $x_2$ as $1-z_2^*$ (using our guess for $z_2^*$), it is possible for the simulator to generate inconsistent double encryptions (with $z_0 \neq z_1$) whenever $\mu[k+1] = z_2 \neq z_2^*$. On the other hand, a decryption key for either $z_0^*$ or $z_1^*$ can be used to implement the final check on the adversary's forgery (since $z_0^* = z_1^*$). These observations enable a Naor-Yung-like double encryption argument to modify the $z_0, z_1$ values in all generated signatures with $\mu[k+1] \neq z_2^*$.
After the above transition is iterated, all signatures are generated with (or checked for) $z_0 := z_1 := \mathbf{RF}_L(\mu)$ for a truly random function $\mathbf{RF}_L$. At this point, we can replace $z_0$ and $z_1$ with $z_0 := z_1 := \mathbf{RF}_L(\mu) + m$ since $\mathbf{RF}_L(\mu)$ is an independently and uniformly random element.
We can replace $z_0 := z_1 := \mathbf{RF}_L(\mu) + m$ with $z_0 := z_1 := x + m$ in a similar way to the case from $\mathbf{RF}_0(\epsilon) = x$ to $\mathbf{RF}_L(\mu)$ (see the proof for the detail). Thus, we can force $\mathcal{A}$ to reuse an $M_i$ in queried messages for $Z_0^*$, as we explained at the beginning of this section.
3.3 Security Proof
Theorem 3.6. If PKE is IND-mCPA-secure and GS is a Groth-Sahai proof system, then xSPS (defined in Section 3.1) is UF-XCMA-secure. Particularly, for all adversaries $\mathcal{A}$, there exist adversaries $\mathcal{B}_1$ and $\mathcal{B}_2$ with running time $T(\mathcal{B}_1) \approx T(\mathcal{A}) \approx T(\mathcal{B}_2)$ and
where L is the smallest even integer that is equal or larger than the bit size of p.
Proof. Let $\mathcal{A}$ be an adversary against UF-XCMA security of xSPS. We prove Theorem 3.6 via Games $G_0-G_3$ defined in Figure 2. We use $\mathrm{Adv}_{G_i}$ to denote the advantage of $\mathcal{A}$ in Game $G_i$.