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I have read about the Fiat-Shamir heuristic for making an interactive proof of knowledge into a non-interactive proof of knowledge. If I apply this to zero-knowledge proof of knowledge, it seems to me that I would destroy the simulator and therefore the zero-knowledgeness.

Does the Fiat-Shamir heuristic destroy zero-knowledgeness? If it does, it it possible to circumvent this?

Link to Fiat-Shamir heuristic.

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In the random-oracle model, it has been shown that the Fiat-Shamir transform preserves the soundness and zero-knowledge of certain families of zero-knowledge (proof of knowledge) protocols (e.g., Sigma protocols) -- see [§5 in BR93,PS00,FKMV12]. Instantiating the random oracle with a concrete hash function is a much harder prospect, but there have been some recent works in this direction as well --- for example, the recent construction of NIZK from LWE relies on such an instantiation of the Fiat-Shamir transform [PS19].

P.S. Zero-knowledge is usually easy to show -- it is arguing soundness that usually causes problems in the non-interactive setting.

[BR93]: Bellare and Rogaway: Random Oracles are Practical: A Paradigm for Designing Efficient Protocols [FKMV12]: Faust et al. On the Non-malleability of the Fiat-Shamir Transform
[PS00]: Pointcheval and Stern: Security Arguments for Digital Signatures and Blind Signatures
[PS19]: Peikert and Sheihan Non-Interactive Zero Knowledge for NP from (Plain) LWE.

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