Gen(par):
crs0, crs1 ⇐ BG(par); For i = 0, 1, 2 : (pki, ski) ⇐ Genp(par) x0 ⇐ Zp; x1 := x2 := 0 ∈ Zp; r0, r1, r2, t0, t1, t2, t3 ⇐ Rc c0 ← Com(crs0, x0; r0); c1 ← Com(crs0, x1; r1); c2 ← Com(crs1, x2; r2) k0 ← Com(crs1, sk0; t0); k1 ← Com(crs1, sk1; t1); k2 ← Com(crs1, sk2; t2); k3 ← Com(crs0, sk0; t3) pk := (crs0, crs1, (ci)0≤i≤2, (ki)0≤i≤3); sk := ((ski, xi, ri)0≤i≤2, (ti)0≤i≤3) Return (pk, sk)
Sign(sk, M ∈ G):
z₀ := z₁ := x₀; z₂ := 0; For i = 0, 1, 2 : ctᵢ ⇐ Enc(pkᵢ, Gzᵢ) ins₀ := (ct₀, M); cv₀ := (c₀, c₁, k₃); w₀ := (x₀, x₁, sk₀); R₀ := (r₀, r₁, t₃) ins₁ := (ctᵢ)₀≤i≤₂; cv₁ := (c₂, (kᵢ)₀≤i≤₂); w₁ := (x₂, (skᵢ)₀≤i≤₂); R₁ := (r₂, (tᵢ)₀≤i≤₂) ρ₀ ⇐ P(crs₀, ins₀, cv₀), (w₀, R₀) //Prove that ins₀ ∈ L₀ and w₀ is committed in cv₀ ρ₁ ⇐ P(crs₁, (ins₁, cv₁), (w₁, R₁)) //Prove that ins₁ ∈ L₁ and w₁ is committed in cv₁ Return σ := (ct₀, ct₁, ct₂, ρ₀, ρ₁)
Ver(pk, M, σ):
Parse σ := ((ctᵢ)₀≤i≤₂, ρ₀, ρ₁) ins₀ := (ct₀, M); cv₀ := (c₀, c₁, k₃); ins₁ := (ctᵢ)₀≤i≤₂; cv₁ := (c₂, k₀, k₁, k₂) Return (V(crs₀, ins₀, cv₀, ρ₀) ∧ V(crs₁, ins₁, cv₁, ρ₁))
Languages:
$L_0 := { (ct_0, M) \mid \exists x_0, x_1 \in \mathbb{Z}p, sk_0 \in SK \text{ s.t.} \ G^{z_0} = G^{x_0}M^{x_1} \land G^{z_0} = \text{Dec}(sk_0, ct_0) }$ $L_1 := { (ct_i){0 \le i \le 2} \mid \exists x_2 \in \mathbb{Z}p, sk_0, sk_1, sk_2 \in SK \text{ s.t.} \ ((z_0 - z_1)(x_2 - z_2) = 0) \land \sum{i=0}^{2} (G^{z_i} = \text{Dec}(sk_i, ct_i)) }$
Figure 1: Our signature scheme xSPS.
Remark 3.1 (Role of proof ρ₀). The main role is to bind a message into a signature. In the real scheme, it is just a proof of the signing key x₀ in ct₀ (and c₀) since x₁ is fixed to 0. Yet the proof is bound to message M through randomness r₁ used for committing to x₁. In the security proof, it can be seen as an encrypted one-time message authentication code (MAC) of M and forces the adversary to reuse given signatures since, intuitively, the adversary cannot generate a new MAC for hidden keys x₀ and x₁.
Remark 3.2 (Role of proof ρ₁). ρ₁ is used for partitioning. It proves that two ciphertexts ct₀ and ct₁ are consistent (namely, the same plaintext is encrypted) or the plaintext in the ciphertext ct₂ is committed to in c₂. In the real scheme, ρ₁ proves the consistency of double encryption ct₀ and ct₁. In the security proof, ρ₁ enables us to achieve two (seemingly incompatible) functionalities under a binding mode CRS. One is forcing the adversary to use consistent ciphertexts in its forgery. A simulator guesses z₂* in the forgery and makes x₂ ≠ z₂* hold. The other is letting the simulator use inconsistent ciphertexts in a special situation achieved using a partitioning technique (see Section 3.2 for more details). In that situation, the simulator can make x₂ = z₂ hold and use a real witness of ρ₀.
Remark 3.3 (On the range of $z_2$). The range of $z_2$ is $\mathbb{Z}_p$ since $z_2$ is the plaintext of $ct_2$. Readers might think we should bind $z_2$ on ${0, 1}$ by using a Groth-Sahai proof since the simulator in the security proof guesses $z_2*$ in the forgery as explained in the previous remark. This is not the case. In fact, even if an adversary uses $z_2*$ such that $z_2* \notin {0, 1}$, it has no advantage because the simualtor uses $x_2$ such that $x_2 \in {0, 1}$ in the security proof. Value $z_2$ affects $ρ_1$. However, to make a valid forgery by using $x_2 = z_2*$ as a witness in $ρ_1$, adversaries have no choice but to use $z_2* \in {0, 1}$ as long as $x_2 \in {0, 1}$. Accordingly, we do not need to bind $z_2$ on ${0, 1}$. This intuition is implemented formally in the proof of Lemma 3.20.
Remark 3.4 (On verifying correctness of pk). Verifying correctness of commitment $k_i$ with respect to $sk_i$ is not necessary for achieving UF-CMA security where keys are generated honestly by definition. But it may have to be verified (once for all at the time of publishing pk) if the scheme is used in an application where signers can be corrupted at the time of key generation.
Remark 3.5 (On XCMA and CMA security of xSPS.). We prove that xSPS is UF-XCMA for efficiency though, in fact, we can prove xSPS is UF-CMA. When we prove UF-CMA, a simulator does not have exponents of queried messages, but the simulator must generate proofs $ρ_0$ for $x_1 \neq 0$ under the binding