I am recently reading about multi party computation and its various existing protocols. From what I understand, all the arithmetic operations are performed over a field or a ring such that when two secret values, a and b, are used to perform secure computation c = f(a,b), the output of the MPC protocol is c mod P (for some modular P).
My question is, in real-life applications when we are performing MPC to get the desired output, this modulo operation may wrap multiple distinct shares into a single field/ring element. Such cases can occur when we are working with real-life applications. In such scenarios, how the accuracy of the original reconstructed output is ensured? The output c may exceed the value P, which will hamper the output as the original accepted output is c, not c mod P.
I am not sure but taking a higher/big P value may mitigate this issue. But in such scenarios what's the issue with directly working on original shares instead of the field elements after the secret sharing is performed?
I am very new to the concept of MPC and may have overlooked some crucial points while understanding the concept. Your help and clarifications are appreciated. Thank you.