SIGN(M) runs $(opk, osk) \leftarrow^{$} Update(par)$ and $\sigma \leftarrow^{$} S(sk, osk, M)$, and then returns $(opk, \sigma)$ to $A$, and records $(opk, M, \sigma)$ to the list $Q_M$.
VER$(opk^*, \sigma^*, M^*)$ returns 1 if there exists $(opk^*, M, \sigma) \in Q_M$ and $M^* \neq M$ and $1 = V(pk, opk^*, M^*, \sigma^*)$, or returns 0, otherwise.
Let POS := (G, Update, S, V) be a structure-preserving partially one-time signature scheme with message space $\mathcal{M}$ and one-time public key space $\mathcal{K}{opk}$, and xSPS := $(\text{Gen}', \text{Sign}', \text{Ver}')$ be a structure-preserving signature scheme with message space $\mathcal{K}{opk}$. The transformed UF-CMA secure SPS scheme, SPS := $(\text{Gen}, \text{Sign}, \text{Ver})$, is defined as follows.
| Gen(par): | Sign(sk, M): | Ver(pk, M, σ): |
|---|---|---|
| (pk1, sk1) ↔$ G(par) | (opk, osk) ↔$ Update(par) | Parse σ = (opk, σ1, σ2) |
| (pk2, sk2) ↔$ Gen'(par) | σ1 ↔$ S(sk1, osk, M) | If V(pk1, opk, M, σ1) = 1 |
| pk := (pk1, pk2) | σ2 ↔$ Sign'(sk2, opk) | ∧Ver'(pk2, opk, σ2) = 1 |
| sk := (sk1, sk2) | Return (opk, σ1, σ2) | then return 1 |
| Return (pk, sk) | Else return 0 |
The correctness and structure-preserving property of SPS are implied by those of POS and xSPS in a straightforward way. The following theorem ([3, Theorem 3]) states UF-CMA security of SPS.
Theorem 2.6. If POS is OT-nCMA secure and xSPS is UF-XRMA secure, then SPS defined as above is UF-CMA secure. In particular, for all adversaries A against UF-CMA security of SPS, there exist adversaries B against OT-nCMA security of POS and C against UF-XRMA security of xSPS with running times $T(\mathcal{A}) \approx T(\mathcal{B}) \approx T(\mathcal{C})$ and $\text{Adv}{\text{SPS}}^{\text{uf-cma}}(\mathcal{A}) \le \text{Adv}{\text{POS}}^{\text{ncma}}(\mathcal{B}) + \text{Adv}_{\text{SPS}}^{\text{uf-xcma}}(\mathcal{C})$.
2.4 Public-Key Encryption Schemes
Definition 2.7 (Public-key encryption). A Public-Key Encryption scheme (PKE) consists of algorithms PKE := $(\text{Gen}_P, \text{Enc}, \text{Dec})$:
The key generation algorithm Gen$_P$(par) takes par $←^{$}$ PGGen$(1^\lambda)$ as input and generates a pair of public and secret keys (pk, sk). Message space $\mathcal{M}$ is implicitly defined by pk.
The encryption algorithm Enc(pk, M) returns a ciphertext ct.
The deterministic decryption algorithm Dec(sk, ct) returns a message M.
(Perfect correctness.) For all par $←^{$}$ PGGen$(1^\lambda)$, (pk, sk) $←^{$}$ Gen$_P$(par), messages $M \in \mathcal{M}$, and ct $←^{$}$ Enc(pk, M), Dec(sk, ct) = M holds.
Definition 2.8 (IND-mCPA Security [10]). A PKE scheme PKE is indistinguishable against multi-instance chosen-plaintext attack (IND-mCPA-secure) if for any $q_e \ge 0$ and for all p.p.t. adversaries A with access to oracle ENC at most $q_e$ times the following advantage function $\text{Adv}_{\text{PKE}}^{\text{mcpa}}(\mathcal{A})$ is negligible,
where $\text{ENC}(M_0, M_1)$ runs $ct^* \leftarrow^{$} \text{Enc}(pk, M_b)$, and returns $ct^*$ to $\mathcal{A}$.
Some public-key encryption schemes, e.g., ElGamal encryption [25] and Linear encryption [17], are structure-preserving and satisfy IND-mCPA security with tight reductions to compact assumptions such as DDH and the Decision Linear assumption [17], respectively (cf. [37]).
2.5 The Groth-Sahai Proof System
We recall the Groth-Sahai proof system and its properties as a commit-and-prove scheme. We follow definitions by Escala and Groth in [27] in a simplified form that is sufficient for our purpose. For a given pairing group par $←^{$} PGGen(1^\lambda)$, the GS-proof system is a non-interactive zero-knowledge proof (NIZK) system for satisfiability of a set of equations over par. Let $\mathcal{L}{par}$ be a family of NP languages defined over par. For a language $\mathcal{L} \in \mathcal{L}{par}$, let $R_\mathcal{L} := {(x, \omega) : x \in \mathcal{L} \text{ and } \omega \in W(x)}$ be a witness relation, where $W(x)$ is the set of witnesses for $x \in \mathcal{L}$. As our construction fixes the language in advance, it is sufficient for our purpose to define the proof system to be specific to $\mathcal{L}$ as follows.