Algorithms for Modular Counting of Roots of Multivariate Polynomials

Abstract

Given a multivariate polynomial P(X1,…,Xn) over a finite field $\ensuremath {\mathbb {F}_{q}}$, let N(P) denote the number of roots over $\ensuremath {\mathbb {F}_{q}}^{n}$. The modular root counting problem is given a modulus r, to determine Nr(P)=N(P)mod r. We study the complexity of computing Nr(P), when the polynomial is given as a sum of monomials. We give an efficient algorithm to compute Nr(P) when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing Nr(P) is ${\rm NP}$-hard. We present some hardness results which imply that our algorithm is essentially optimal for prime fields. We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials.