Common use of There exists a PPT algorithm MS Clause in Contracts

There exists a PPT algorithm MS. MVerifyInv that on input the public parameters ppms, a message m and a multi-signature σms, outputs vk ∈ R. { } ∈∈ We require that for vkagg = ƒagg( vki i S) R and MS.Verifyagg-key(ppms, vkagg, m, σms) = 1, it holds that MS.MVerifyInv computes the corresponding unique and well-defined key vkagg, i.e.,

There exists a PPT algorithm MS. MVerifyInv that on input the public parameters ppms, a message m and a multi-signature σms, outputs vk ∈ R. { } ∈∈ We require that for vkagg = ƒagg( vki i S▇▇▇▇({vki}i∈S) ∈ R and MS.Verifyagg-key(ppms, vkagg, m, σms) = 1, it holds that MS.MVerifyInv computes the corresponding unique and well-defined key vkagg, i.e.,

There exists a PPT algorithm MS. MVerifyInv that on input the public parameters ppms, a message m and a multi-signature σms, outputs vk ∈ R. { } ∈∈ We require that for vkagg = ƒagg▇▇▇▇( vki i S) R and MS.Verifyagg-key(ppms, vkagg, m, σms) = 1, it holds that MS.MVerifyInv computes the corresponding unique and well-defined key vkagg, i.e.,