I need to know if there is an efficient algorithm for knowing the following problem?
Consider a monic polynomial $f$ of degree $3$ on the finite field $F_q$ when $q$ is the power of a prime number. Suppose S is the set of values of $f$ for all elements of $F_q$.
$\begin{gathered} f\left( x \right) = {x^3} + a{x^2} + bx + c \hfill \\ \hfill \\ S = \left\{ {f\left( x \right)|x \in {F_q}\,} \right\} \hfill \\ \end{gathered} $
An algorithm that tells me in good time that the number of squares S in Fq is greater or the number of not-squares S.
That is, in fact, the output of the algorithm is true or false