I am implementing Shamir's secret sharing scheme on arbitrary binary files. I don't intend to use this; this is a project to help me explore cryptography. In setting up the finite field arithmetic, I chose my field to be GF(257).
My understanding is that a prime GF is needed to ensure that a single modular inverse exists for each value in the field (this conflicts with the answer to How can Shamir's method for secret sharing work in the GF(256)?).
Can someone explain this discrepancy?