The challenge c in Sigma protocol (using Fiat Shamir)

As is known to all, the following picture depicts a sigma protocol, and to eliminate the interactivity, Prover can generate c by hashing (t, y) using Fiat Shamir transform. My question is: 1). Can c be obtained as hash(0x1234, 0x4321) by the Prover, (where 0x1234, 0x4321 are public parameters at setup)? 2). If above answer is yes, can a same c be used by different Provers at the same time in Fiat Shamir transform?

In the Fiat-Shamir transform, the challenge $$c$$ would be obtained by hashing the commitment $$t$$. The intuition is that in sigma protocols there is only one $$c$$ (or a negligible fraction, depending on who you ask) per $$t$$ that will allow the prover to convince the verifier of a false claim. Now if the hash function is perfect, each challenge $$H(t)$$ generated by the prover will be random and will fail to convince the verifier with high probability. When you hash public parameters, the challenge is the same at every execution, so the prover can choose $$t$$ based on this challenge.