Would Evaluating a polynomial at uniformly random points outputs random values? I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?

A fixed polynomial over a finite field$~F$ will rarely transform uniformly chosen inputs into uniformly chosen outputs: this will happen if and only if the polynomial function $F\to F$ it defines is a bijection. This can certainly happen (every function $F\to F$ is a polynomial function, and uniquely so if one limits the degree to be less than $|F|$; moreover every polynomial of degree$~1$ defines a bijective polynomial function), but as $|F|$ increases this is a vanishing proportion of all functions, and if the sport is to permute the elements of your field, it is not clear wat advantage there is in describing that permutation by a polynomial.
