Counting Value Sets: Algorithm and Complexity

Let p be a prime. Given a polynomial in Fpm [x] of degree d over the finite field Fpm , one can view it as a map from Fpm to Fpm , and examine the image of this map, also known as the value set. In this paper, we present the first non-trivial algorithm and the first complexity result on explicitly computing the cardinality of this value set. We show an elementary connection between this cardinality and the number of points on a family of varieties in affine space. We then apply Lauder and Wan’s p-adic point-counting algorithm to count these points, resulting in a non-trivial algorithm for calculating the cardinality of the value set. The running time of our algorithm is (pmd)O(d). In particular, this is a polynomial-time algorithm for fixed d if p is reasonably small. We also show that the problem is #P-hard when the polynomial is given in a sparse representation, p = 2, and m is allowed to vary, or when the polynomial is given as a straight-line program, m = 1 and p is allowed to vary. Additionally, we prove that it is NP-hard to decide whether a polynomial represented by a straight-line program has a root in a prime-order finite field, thus resolving an open problem proposed by Kaltofen and Koiran in [5, 6].