Why are the factors p and q (for n) secret in Fiat-Shamir Authentication? I am studying about this authentication and I don't understand why the factors of $n$ are kept secret as I can't see how they would give the prover or verifier anymore information.

My professor said that the verifier could learn the secret of the prover with the factors, but I don't understand how he would come about that mathematically. An explanation would be great to learn and understand this: Why are the factors $p$ and $q$ (for $n$) secret in Fiat-Shamir Authentication?
Knowledge of $p$ and $q$ makes computing square roots modulo $n=pq$ easy.

Using the Tonelli-Shanks algorithm, one may determine square roots modulo $p$ and $q$ individually and then combine them to a square root modulo $n$ using the Chinese remainder theorem.

Therefore, anyone who knows $p$ and $q$ can compute the prover's private key (which is a square root of the public key) to impersonate them.