Output-sensitive Sparse Polynomial GCD over Finite Fields is NP-hard

2026-06-10

AI Summary

This study addresses the computational complexity of output-sensitive greatest common divisor (GCD) computation for sparse univariate polynomials over finite fields. By leveraging the algebraic structure of finite fields and employing BPP many-one reductions, the authors establish for the first time that this problem is NP-hard, thereby resolving Challenge 5 from “The Sparsity Challenges.” They further demonstrate that the problem of detecting roots of unity is also NP-hard. These results imply that, under standard complexity-theoretic assumptions, no polynomial-time randomized algorithm in BPP can solve the sparse polynomial GCD problem over finite fields. Consequently, the work rigorously establishes the intrinsic computational hardness of this fundamental task in symbolic computation and provides critical theoretical limits with implications for cryptography.

Abstract

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.

Problem

• Sparse Polynomial GCD
• Finite Fields
• NP-hard
• Output-sensitive
• Roots of Unity Detection

Innovation

• sparse polynomial GCD
• finite fields
• NP-hard
• output-sensitive
• roots of unity detection