# ZKP for product of two primes

I'm struggling to understand the intuition of the zero knowledge-ness of this proof from the following paper.

The proof is a 2 round where the verifier asks the prover to extract square roots of quadratic residues.

I've read the HVZK proof (shown in the picture below) and I agree that you can produce a transcript that distributes like a normal proof, however I still can't get over the simple fact that:

• Without the witness (factorization) you can't extract a square root modulo $$N$$.
• When the proof is turned into a NIZK via the Fiat-Shamir transformation, the prover records square roots in their raw form inside the proof, so the verifier learns square roots of numbers in the group which it could not compute on its own before.

Isn't that a "knowledge" leak? Why can this proof be turned into a NIZK? "HVZK" stands for "honest-verifier zero knowledge", right? Your objection is that a dishonest verifier can choose a random $$x_i$$, compute $$\rho_i\equiv x_i^2\mod N$$, and then when they get $$\sigma_i$$ from the prover, there is a $$1/4$$ chance that $$gcd(\sigma_i-x_i,N)$$ is a non-trivial factor. But an honest verifier should generate random $$\rho_i$$.
This should be secure, based on the following: Suppose $$A$$ is an algorithm that can factor $$N$$ given random $$\rho_i$$ and $$\sigma_i$$ such that $$\sigma_i^2\equiv \rho_i\mod N$$. Then we can factor any $$N$$ with $$A$$ by picking a random $$\sigma_i$$, computing $$\rho_i\equiv\sigma_i^2\mod N$$, and sending $$\rho_i$$ and $$\sigma_i$$ to $$A$$.
• Thanks for your comment, I understand the difference between HVZK and regular ZK and Fiat Shamir heuristic, it's just that this protocol "feels" odd in that sense: Before observing the ZKP, I cannot root square mod N any number. However after observing the ZKP now I know several numbers of the form $(x, \sqrt{x}~mod~N)$ however I could never have computed these roots before observing the ZKP, so how is zero knowledge preserved here if I can compute new things I wasn't able to before? – yacovm Jun 19 at 18:34