Sublinear Root Detection and New Hardness Results for Sparse Polynomials over Finite Fields

Abstract

We present a deterministic $2^{O(t)}q^{\frac{t-2}{t-1}+o(1)}$ algorithm to decide whether a univariate polynomial $f$, with $t$ monomial terms and degree $<q$, has a root in the finite field $\mathbb{F}_q$. Our method is the first with complexity sublinear in $q$ when $t$ is fixed. We also prove a structural property for the nonzero roots in $\mathbb{F}_q$ of any $t$-nomial: The nonzero roots always admit a partition into no more than $2(q-1)^{\frac{t-2}{t-1}}$ cosets, each associated with one of two subgroups $S_1\subseteq S_2$ of $\mathbb{F}^*_q$. This can be thought of as a finite field analogue of Descartes' rule. A corollary of our results is the first deterministic sublinear algorithm for detecting common degree-one factors of $k$-tuples of $t$-nomials in $\mathbb{F}_q[x]$ when $k$ and $t$ are fixed. When $t$ is not fixed we show that, for $p$ prime, detecting roots in $\mathbb{F}_p$ for $f$ is $\mathbf{NP}$-hard with respect to $\mathbf{BPP}$-reductions. Finally, we prove that if the complexity of root detection is sublinear (in a refined sense), relative to the straight-line program encoding, then $\mathbf{NEXP}\nsubseteq\mathbf{P/poly}$.