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We use Fiat–Shamir heuristic to turn Schnorr ZK to NIZK. So now NIZK is secure against chosen message attacks in the random oracle model, that is, ideal hash function: every unique query leads to a truly random response.

Ok, I get why our Hash function should not be reversible - otherwise, we disclose skey.

But what is the danger of collisions?

Suppose, a prover computes NIZK proof (r,c):

c = Hash( g || V || pkey || User Id || Other Info )

r = v + skey*c mod q

V = g^v mod p, v - random

Even if I find any other combination that results in c, it's unlikely I can forge proof.

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First, I do not think reversing the hash function will help you disclose skey as nothing in the Hash needs to be private. The Hash in Schnorr NIZK is a practical instantiation of the random oracle, not perfectly though. Here the random oracle is used to commit or sign V, so that you can not choose V based on c. Note that it is trivial to get V given y, c, g, r, and pass the verification $Vy^c = g^r$.

As for the danger of collisions, it may not be very relevant to the security proof. Hash functions are just how we implement random oracles in practice. I think a better way to look at this is as follows. If one can forge a proof, then c should be equal to the output of the random oracle with input (g, V, pkey, ...). This happens with small probability, as a random oracle always outputs a purely random value given a new input.

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  • $\begingroup$ You are right, all values in the Hash are public - there is no danger for skey. $\endgroup$ – pintor Jan 25 '18 at 9:32

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