Counting Value Sets: Algorithm and Complexity

TL;DR

This paper introduces an algorithm to compute the size of the value set of polynomials over finite fields, analyzes its complexity, and establishes hardness results for related problems, including root detection.

Contribution

It presents the first non-trivial algorithm for counting value set cardinality and proves complexity bounds and hardness results for related polynomial problems.

Findings

Algorithm for counting value set size with runtime (pmd)^{O(d)}

Polynomial-time algorithm for fixed degree d when p is small

NP-hardness of root detection in certain polynomial representations

Abstract

Let $p$ be a prime. Given a polynomial in $\F_{p^m}[x]$ of degree $d$ over the finite field $\F_{p^m}$, one can view it as a map from $\F_{p^m}$ to $\F_{p^m}$, 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 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 \cite{Kaltofen03,KaltofenKo05}.