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Table 1 Games in the security proof

From: Predicate encryption against master-key tampering attacks

Gres: Replace the restriction RN(X,Y)=0 with \(R_{p_{2}}(X, Y^*) = 0\).
G0: \((\text {MSK}, \text {PP}, g_{2}, \hat {\mathbf {h}})\leftarrow \mathbf {SetupSF}(\lambda)\)
  \(\text {CT}_{Y}^{*} \leftarrow \mathbf {EncryptSF}\left (\text {PP}, Y^{*}, \textit {M}_{b}, g_{2}, \hat {\mathbf {h}}\right) \)
Gk,1: \(\hat {\alpha }_{j} \xleftarrow {\$} \mathbb {Z}_{N}, \text {SK}_{j} \leftarrow \left \{\begin {array}{ll} \mathbf {KeyGenSF}\left (\phi _{i}(\alpha), X_{j}, g_{2}, {\mathbf 0}, 3, \hat {\alpha }_{j}\right) & \text {if } j < k \\ \mathbf {KeyGenSF}\left (\phi _{i}(\alpha), X_{j}, g_{2}, \hat {\mathbf {h}}, 1, 0\right) & \text {if } j = k \\ \mathbf {KeyGen}\left (\phi _{i}(\alpha), X_{j}\right) & \text {if } j > k \\ \end {array}\right.\)
Gk,2: \(\hat {\alpha }_{j} \xleftarrow {\$} \mathbb {Z}_{N}, \text {SK}_{j} \leftarrow \left \{\begin {array}{ll} \mathbf {KeyGenSF}\left (\phi _{i}(\alpha), X_{j}, g_{2}, \mathbf {0}, 3, \hat {\alpha }_{j}\right) & \text {if } j < k \\ \mathbf {KeyGenSF}\left (\phi _{i}(\alpha), X_{j}, g_{2}, \hat {\mathbf {h}}, 2, \hat {\alpha }_{j}\right) & \text {if } j = k \\ \mathbf {KeyGen}\left (\phi _{i}(\alpha), X_{j}\right) & \text {if } j > k \\ \end {array}\right.\)
Gk,3: \(\hat {\alpha }_{j} \xleftarrow {\$} \mathbb {Z}_{N}, \text {SK}_{j} \leftarrow \left \{\begin {array}{ll} \mathbf {KeyGenSF}\left (\phi _{i}(\alpha), X_{j}, g_{2}, \mathbf {0}, 3, \hat {\alpha }_{j}\right) & \text {if } j \le k\\ \mathbf {KeyGen}\left (\phi _{i}(\alpha), X_{j}\right) & \text {if } j > k\\ \end {array}\right.\)
\(\mathbf {G}_{q_{1}+1}\): \(\text {SK}_{j} \leftarrow \mathbf {KeyGenSF}(\phi _{i}(\alpha), X_{j}, g_{2}, \hat {\mathbf {h}}, 1, 0) \)
\(\mathbf {G}_{q_{1}+2}\): \(\hat {\alpha } \xleftarrow {\$} \mathbb {Z}_{N} \)
  \(\text {SK}_{j} \leftarrow \mathbf {KeyGenSF}\left (\phi _{i}(\alpha), X_{j}, g_{2}, \hat {\mathbf {h}}, 2, \hat {\alpha }\right) \)
\(\mathbf {G}_{q_{1}+3}\): \(\text {SK}_{j} \leftarrow \mathbf {KeyGenSF}\left (\phi _{i}(\alpha), X_{j}, g_{2}, \mathbf {0}, 3, \hat {\alpha }\right) \)
Gfinal: \(M \xleftarrow {\$} \mathcal {M}, \text {CT}_{Y}^{*} \leftarrow \mathbf {EncryptSF}\left (\text {PP}, Y^{*}, \textit {M}, g_{2}, \hat {\mathbf {h}}\right) \)