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When I study a zk-SNARK scheme, the scheme claims to be transparent. Does this mean that this scheme does not require a trusted setup? Furthermore, if a NIZK scheme includes a Common Reference String (CRS), does that scheme still remain transparent?

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Exactly, a scheme that is transparent does not require a trusted setup. These are typically interactive schemes that are converted to non-interactive ones using the Fiat-Shamir heuristic (and thus rely on the random oracle model).

A Common Reference String (CRS), often also called a Structured Reference String (SRS), is the output of a trusted setup with a specific form and is the common input to all the parties, i.e., the prover and the verifier. Such schemes are not transparent as they require a trusted setup.

There are various approaches to overcome the problem of having a trusted party that generates the CRS/SRS. In particular, schemes that consider that the computation of the CRS can be subverted (i.e., one obtains guarantees even if the CRS generator is malicious) or schemes with an updatable CRS. In the latter setting anyone can update the CRS and provide "update proofs" that let everyone check that the update was performed correctly. If one, either the generator or any of the updaters are honest, then one can have "trust" in the CRS. These two approaches thus allow to reduce the trust required in the CRS generation.

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  • $\begingroup$ It seems that the word "transparent" is often used in NIZK without trusted setup. If a scheme from another field, such as ring signatures, is without trusted setup, can we use the term "transparent" to describe this ring signature scheme? $\endgroup$
    – user109993
    Oct 9, 2023 at 14:13
  • $\begingroup$ I have never seen it in such a context, but I guess you could. $\endgroup$
    – DrLecter
    Nov 21, 2023 at 15:32

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