i would like to know why there is a problem of not using random x-coordinates in shamir secret sharing schemes.
I consider that after evaluating the points in a polynomial $f(x)$, the share is composed by: $(x, f(x))$, where the $f(x)$ is secret and the $x$ could be public. So, why i always find in some literature that it is a problem using not random x coordinates? Could i have an example of why is it a problem?
Associating the shamir scheme with ortogonal arrays (respecting the strength and lambda properties), i know that fixing the x-coordinates i am restricting the matrix to only the known columns and, consequently, we have a smaller matrix. But we still have the perfect privacy properties, because even knowing the x-coordinates, each secret appears exactly the same quantity of times for each t-tuple, where $t$ is the threshold. Therefore, even knowing the x-coordinates, i can not know anything about the secret.