I am trying to write an assay about Non Interactive Zero-Knowledge proofs and would like to take the simple discrete logarithm problem example fallowing the Feige-Fiat-Shamir heuristics.

I understand the interactive steps are:

1. P wants to proof he knows x where y = g^x
2. P picks a random v and computes t = g^v and sends t to V
3. V picks a random c and sends it to P
4. P computes r = v - cx and sends it to V
5. V checks t = g^r * y^c

while the non-interactive steps would be like this:

1. P wants to proof he knows x where y = g^x
2. P picks a random v and computes t = g^v and keeps it as first term of the proof
3. P computes c = H(g,y,t)
4. P computes r = v - cx and keeps it as second term of the proof
5. V checks t = g^r * y^H(g,y,t)

Now I see V doesn't actually get any hint about the value of x P wants to proof he knows whithout revealing.

But the definition for Zero-Knowledge should be a Simulator could compute the whole proof in probabilistic polynomial time.

How can this definition be applied to the non-interactive procedure? How can a simulator repoduce the proof (calculating the correct hashes) if he does not know x, thus he can't compute r which is part of t which in turn is an input of H?

Can anybody link me some simple explanation of this?

Thanks