Shamir Scheme: Whats the problem of not using random x-coordinates? 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.
i would like to know why there is a problem of not using random x-coordinates in shamir secret sharing schemes.


There is no such problem.

The only requirements on the $x$-coordinates are:


They're all in the range $(1, p-1)$ (or $p^k - 1$ if you're using the extension field $GF(p^k)$; you can't use 0 as a coordinate (as $f(0)$ is the secret), and you can't you values outside the range.
They're distinct (if they're not, then the two shares with the same $x$ coordinate are the same share).


ANy arbitrary set of $x$-coordinates that meet these two requirements work just fine.


  So, why i always find in some literature that it is a problem using not random x coordinates?


I suspect that you are misinterpreting the literature.  For the coefficients of the secret polynomial $f(x)$, it is necessary that the non-constant terms be random; if they are not, then it is possible for a small group to recover the secret.

There is no corresponding requirement on the $x$; they can be any values you find convenient (as long as they meet the above two requirement)