Oracle $$\mathcal {O}_{\pi }$$: {\begin {aligned} \mathcal {O}_{\pi }(\textsf {CRS}_{i}) = \left \{\begin {array}{ll} \pi _{i} & \text {if there exists unique}\ \pi _{i} \text {stored in oracle}\ \mathcal {O}_{\pi }\ \text {and} \\ & \textsf {PC}.\textsf {V}_{\textsf {cert}}(1^{n},\textsf {CRS}_{i},\pi _{i}) = 1 \\ \perp & \text {otherwise} \end {array}\right. \end {aligned}} Circuit $$\mathcal {P}^{n,c_{3},\textsf {{PP}},\textsf {{K}},\rho _{\textsf {{CRSGen}}}}$$: {\begin {aligned} \mathcal {P}(d,\rho _{3}) \,=\, \left \{\begin {array}{ll} \kappa & \text {if}\ c_{3}\,=\,\textsf {Com}(d,\rho _{3}),\ \text {then set}\ \kappa \! :=\!\textsf {PC}.\textsf {CRSGen}(\textsf {PP},\textsf {K},d,\rho _{\textsf {{CRSGen}}})\\ \perp & \text {otherwise} \end {array}\right. \end {aligned}} Equivalent Circuit $$\mathcal {Q}^{n,c_{3},\kappa }$$: $$\mathcal {Q}(d,\rho _{3}) = \left \{\begin {array}{ll} \kappa &\text {if}\ c_{3}=\textsf {Com}(d,\rho _{3})\\ \perp & \text {otherwise} \end {array}\right.$$ Language Λ1: We say (h,c1,c2,r)∈Λ1, iff there exist $$(M,\rho _{1},\mathcal {O}_{\pi },(j,s_{j}),\rho _{2})$$ such that – $$\rho _{1},\rho _{2},s_{j} \in \{0,1\}\text {} ^{n}, j \in [m], M, \mathcal {O}_{\pi } \in \{0, 1\}\text {} ^{n^{\log \log n}}$$; – c1=Com(h(M),ρ1); – c2=Com(h(q),ρ2) where $$\textsf {q}=((M,\mathcal {O}_{\pi }),(j,s_{j}),r)$$. Language Λ2: We say (h,PP,c2,c3,r)∈Λ2, iff there exist $$(M,\mathcal {O}_{\pi },(j,s_{j}),d,\rho _{2},\rho _{3})$$ such that – $$d,s_{j},\rho _{2},\rho _{3} \in \{0, 1\}\text {} ^{n}, j \in [m], M, \mathcal {O}_{\pi } \in \{0, 1\}\text {} ^{n^{\log \log n}}$$; – c2=Com(h(q),ρ2) where $$\textsf {q}=((M,\mathcal {O}_{\pi }),(j,s_{j}),r)$$; – c3=Com(d,ρ3) where d=PC.PreGen(PP,q). Language Λ3: We say $$(\textsf {PP},c_{3},\hat {\mathcal {P}}_{\textsf {{CRSGen}}}) \!\in \! \Lambda$$3, iff there exist $$\left (\textsf {K},\mathcal {P}^{c_{3},\textsf {{PP}},\textsf {{K}},\rho _{\textsf {{CRSGen}}}},\rho _{\textsf {{Setup}}}, \rho _{\textsf {{CRSGen}}},\rho _{i\mathcal {O}}\right)$$ such that – $$\textsf {K},\rho _{\textsf {{Setup}}},\rho _{\textsf {{CRSGen}}},\rho _{i\mathcal {O}} \in \{0, 1\}\text {} ^{n}$$; – (PP,K)=PC.Setup(1n,D,ρSetup); – $$\hat {\mathcal {P}}_{\textsf {{CRSGen}}}=i\mathcal {O}\left (\mathcal {P}^{c_{3},\textsf {{PP}},\textsf {{K}},\rho _{\textsf {{CRSGen}}}},\rho _{i\mathcal {O}}\right)$$. Language Λ4: We say $$(\textsf {PP},\hat {\mathcal {P}}_{\textsf {{CRSGen}}},c_{5}) \in \Lambda$$4, iff there exist (d,ρ3,π) such that – d,ρ3,π∈{0,1}n; – $$\kappa = \hat {\mathcal {P}}_{\textsf {{CRSGen}}}(d,\rho _{3})$$; – c5=Com((PP,κ),ρ5); – PC.Vcert(1n,(PP,κ),π)=1.