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Let us therefore turn to the T (n)-simulatability property. On a high level the proof follows the structure of the sequential composition lemma for ZK proofs of Goldreich and Oren . We start by “partitioning” the malicious verifier VQ∗ into Q(n) phases, each of which is the execution of a verifier for a “stand-alone” interactive proof (P, V ), called V ∗ . The new stand-alone verifier V ∗ will communicate Q(n) times with a real prover P , and we wish to show that V ∗ does not “learn” anything from these Q(n) interactions.
We note that the proof of the fact that ZK implies WH in models with shared object is a straight-forward adaptation of the proof of this fact for the plain model . Interestingly, concerning WI, on the other hand, such a simple adaptation can no longer be done. This was shown in  already for the case of non-interactive ZK in the CRS model. We mention that reason for this problem stems from the fact that the definition of ZK in models with shared object allows the simulator to “choose” the shared object, when performing the simulation.
It remains to show that the output of the simulator MQ is indistinguishable from the output of the verifier VQ in a real interaction with a prover. , HQ(n) . , for i = 1 to j let zi = M (x, zi−1 ). , for i = j to Q(n) let zi = P, V ∗ (zi−1 ) (x). • Output zQ(n) . The claim consists of proving that hybrids H0 and HQ(n) are computationally indistinguishable. Assume for contradiction that this is not the case. By a standard argument this implies that there are two consecutive hybrids distributions Hk , Hk+1 that are distinguishable.