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I was wondering if we could use any general symmetric key MAC, a collision resistant hash $H$ and zk-SNARK (or any general NIZK scheme) to construct a signature scheme. I have a MAC key $k$, which is also my private key, and the public key is $h = H(k)$. As a signature for the message is $(mac = MAC_k(m),\pi)$ where $\pi$ refers to a proof using SNARK circuit which takes public input $m,h,mac$ and private input (witness) $k$ and calculates $mac' = MAC_k (m), h' =H(k)$ and outputs $1$ iff $mac=mac'$ and $h'=h$ .

Looks like we need to be extra careful if the MAC above is an HMAC with the same hash function $H$ since it uses actual $Key' = H(Key)$ if the $Key$ longer than block size,other than that are there other issues with it? Is it secure?

If it is not secure can anyone give a reason why? Way to recover key or forge a signature would be even better.

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This is a valid paradigm for building a signature scheme, although a secure commitment scheme should be used to commit to $k$ instead of just using $H(k)$ as the public key. This type of construction was published by Bellare and Goldwasser at CRYPTO'89; see New Paradigms for Digital Signatures and Message Authentication Based on Non-Interactive Zero Knowledge Proofs. Formally, they use a NIZK, as you stated.

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