We introduce weaker models for non-interactive zero knowledge, in which the dealer is not restricted to deal a truly random string and may also have access to the input to the protocol (i.e. the statement to prove). We show in these models a non-interactive statistical zero-knowledge proof for every language that has (interactive) statistical zero-knowledge proof, and a computational zero-knowledge proof for every language in. We also show how to change the latter proof system to fit the model of non-interactive computational zero-knowledge with preprocessing to improve existing results in term of the number of bit commitments that are required for the protocol to work.
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