any key pairs $(pk, sk) \stackrel{\mathcal{S}}{\leftarrow} \text{Gen}P(\text{par})$ and $(pk', sk') \stackrel{\mathcal{S}}{\leftarrow} \text{Gen}P(\text{par})$, any messages $m$ and $m'$ in the legitimate message space, and any randomness $s$, it holds that $\text{Enc}(pk', m'; s) = \text{SimEnc}(sk', m', \text{Enc}(pk, m; s))$. In [11], Bellare et al. formally defined such a property as reproducibility. Given reproducible PKE and its ciphertext $ct_b \stackrel{\mathcal{S}}{\leftarrow} \text{Enc}(pk_b, G_1^{z_b}; s)$, we can compute another ciphertext $ct{1-b} \stackrel{\mathcal{S}}{\leftarrow} \text{SimEnc}(sk{1-b}, G_1^{z_{1-b}}, ct_b)$ without knowing $sk_b$ or $s$. All reduction steps with respect to the CPA security of PKE go through using $\text{SimEnc}$ and simulated GS proofs. Precisely, we use $\text{SimEnc}$ in Lemma 3.21 to compute $ct_0$ from given $ct_1$. Similar adjustment applies to Lemma 3.23, 3.35 and 3.37.
As shown in [11], ElGamal encryption (EG) is reproducible. Let $(y, G_1^y)$ and $(y', G_1^{y'}) \in \mathbb{Z}_p \times \mathbb{G}_1$ be two key pairs of ElGamal encryption. Given ciphertext $(M \cdot (G_1^y)^s, G_1^s)$ of message $M$ with $s$ and public key $G_1^y$, one can compute $(M' \cdot (G_1^s)^{y'}, G_1^s)$ for any $M'$ using secret key $y'$. It is exactly the same ciphertext obtained from the regular encryption with common randomness $s$. We thus encrypt $z_0$ and $z_1$ with ElGamal encryption in $\mathbb{G}_1$ using the same randomness and removing redundant $G_1^s$. For encrypting $z_2$, we also use ElGamal but in $\mathbb{G}_2$. Bellare et al. show that the multi-message chosen-plaintext security for each encryption holds under the DDH assumption in respective groups, which is directly implied by the SXDH assumption [10]. We thus have:
Theorem 4.1. For all adversaries $\mathcal{A}$ against IND-mCPA security of EG, there exists an adversary $\mathcal{C}$ against the SXDH assumption with running time $T(\mathcal{C}) \approx T(\mathcal{A})$ and $\text{Adv}{\text{PKE}}^{\text{mcpa}}(\mathcal{A}) \le 2 \text{Adv}{\text{PGGen}}^{\text{sxdh}}(\mathcal{C}) + \frac{1}{p}$.
4.2 Concrete Scheme for Unilateral Messages
We present a concrete scheme, SPSu1, for signing messages in $\mathbb{G}_1$. We use a structure-preserving one-time signature scheme, POSu1, taken from the results of Abe et al. [3], and the SXDH-based instantiation of GS proof system. The description of POSu1 is blended into the description of SPSu1. For the GS proofs, however, we only show concrete relations in this section and present details of computation in Section 4.4.
We use notations $[x]_i$ and $[\tilde{x}]_1$ as a shorthand of $\text{Com}(\text{crs}_i, x)$ and $\text{Com}(\tilde{\text{crs}}_1, x)$, respectively. We abuse these notations to present witnesses in a relation. It is indeed useful to keep track which CRS and which source group is used to commit to a witness. This notational convention is used in the rest of the paper.
Scheme SPSu1: Let par := $(p, \mathbb{G}_1, \mathbb{G}_2, \mathbb{G}_T, e, G, \tilde{G})$ be a description of Type-III bilinear groups generated by PGGen$(1^\lambda)$.
SPSu1.Gen(par). Generates $\text{crs}_0$, and $(\text{crs}_1, \tilde{\text{crs}}_1)$ as shown in (18). Picks $x_0 \stackrel{\mathcal{S}}{\leftarrow} \mathbb{Z}_p$ and set $x_1 = x_2 := 0$. Generates three ElGamal keys $\tilde{Y}_0 := \tilde{G}^{y_0}, \tilde{Y}_1 := \tilde{G}^{y_1}$, and $Y_2 := G^{y_2}$ where $y_i \stackrel{\mathcal{S}}{\leftarrow} \mathbb{Z}_p$ for $i = 0, 1, 2$. Then computes commitments
as shown in Equation (19). Generates a persistent key pair of POSu1 by $w \stackrel{\mathcal{S}}{\leftarrow} \mathbb{Z}_p^*$, $\gamma_i \stackrel{\mathcal{S}}{\leftarrow} \mathbb{Z}_p^*$, $\tilde{G}_r := \tilde{G}^w$, and $\tilde{G}_i := \tilde{G}_r^{\gamma_i}$ for $i = 1, \dots, n_1$. Outputs pk and sk defined as $pk := (G, \tilde{G}, \text{crs}_0, \text{crs}_1, \tilde{\text{crs}}_1, \tilde{Y}_0, \tilde{Y}_1, Y_2, [x_0]_0$, $[x_1]_0$, $[\tilde{x}_2]1$, $[y_0]0$, $[y_0]1$, $[y_1]1$, $[\tilde{y}2]1$, $\tilde{G}r, \tilde{G}1, \dots, \tilde{G}{n_1})$, and $sk := (x_0, y_0, y_1, y_2, r{x_0}, r{x_1}, r{x_2}, r{y_0}, r{y_1}, r{y_2}, w, \gamma_1, \dots, \gamma{n_1})$, where par and pk are implicitly included in pk and sk, respectively.
SPSu1.Sign(sk, M). Given sk as defined above and $M := (M_1, \dots, M_{n_1}) \in \mathbb{G}_1^{n_1}$, proceeds as follows.
- Generate one-time POSu1 key pair $\alpha \stackrel{\mathcal{S}}{\leftarrow} \mathbb{Z}_p^*$ and $\tilde{A} := \tilde{G}^\alpha$, and compute a one-time signature, $(Z, R)$, by
where $w, \gamma_1, \dots, \gamma_{n_1}$ are taken from sk, and $\rho$ is chosen uniformly from $\mathbb{Z}_p$.
- Encrypt $z_0 = z_1 := x_0$, and $z_2 := 0$ as $(\tilde{E}{z_0}, \tilde{E}{z_1}, \tilde{E}_s) := (\tilde{G}^{z_0}\tilde{Y}_0^s, \tilde{G}^{z_1}\tilde{Y}1^s, \tilde{G}^s)$ and $(E{z_2}, E_t) := (G^{z_2}Y_t^t, G^t)$, where $s,t \stackrel{\mathcal{S}}{\leftarrow} \mathbb{Z}_p$.