Do you know why an integer implementation is insecure as stated on the Wikipedia page? I built a toy implementation in Mathematica but I can't see any way one could recover the constant term with fewer than k-1 points on the line.
I'm not sure, but I believe it has to do with the infinite extent of integers. I'm guessing that if you did it only with integers modulo a large prime number, it would be more secure. I encourage you to read up on it
