In this paper, we prove that output-sensitive sparse polynomial GCD computation over finite fields is NP-hard under BPP many-one reduction. More precisely, for two sparse univariate polynomials $f,g$ with finite field coefficients, there exists no randomized algorithm to compute $\mathrm{gcd}(f,g)$, which is polynomial-time in the sizes of $f,g,\gcd(f,g)$ under the standard complexity assumption $\mathrm{NP}\nsubseteq\mathrm{BPP}$. This settles the open problem posed as Challenge 5 in The Sparsity Challenges in the finite field setting. Furthermore, we show that the Roots of Unity Detection problem over finite fields is NP-hard; that is, determining whether the GCD of a sparse univariate polynomial and $x^n - 1$ has nonzero degree is NP-hard.

翻译：本文证明了，在BPP多对一归约下，有限域上的输出敏感稀疏多项式最大公因式（GCD）计算问题是NP难的。更精确地说，对于两个具有有限域系数的稀疏一元多项式$f,g$，在标准复杂性假设$\mathrm{NP}\nsubseteq\mathrm{BPP}$下，不存在一个计算$\mathrm{gcd}(f,g)$的随机化算法，其时间复杂度是$f,g,\gcd(f,g)$大小的多项式。这解决了有限域背景下“稀疏性挑战”系列问题中挑战5的公开问题。此外，我们证明了有限域上的单位根检测问题是NP难的；即判定一个稀疏一元多项式与$x^n - 1$的GCD是否具有非零次数是NP难的。