{
  "abstract": "Finding the roots of a polynomial over a finite field $\\mathbb F_q$ in deterministic polynomial time is a long-standing open problem. Classical deterministic methods work in the multiplicative group $\\mathbb F_q^\\times$ and are efficient only when $q-1$ is smooth, which is rarely the case. We propose a method that replaces $\\mathbb F_q^\\times$ by the groups of rational points of an elliptic curve $E$ and of its quadratic twist $\\widetilde E$. The computation splits into an offline phase, which depends only on $\\mathbb F_q$ and outputs a short \\emph{elliptic preprocessing datum} $\\mathcal D$, and an online phase, which processes the input polynomial. Writing $Ξ(\\mathcal D)$ for the largest prime factor of $\\lvert E(\\mathbb F_q)\\rvert\\cdot\\lvert\\widetilde E(\\mathbb F_q)\\rvert$, the online phase is deterministic and finds the roots of any $f\\in\\mathbb F_q[X]$ of degree $n$ within $\\widetilde O(Ξ(\\mathcal D)^2\\,n\\log^2 q+n^ω\\log q+n\\log^3 q)$ operations in $\\mathbb F_q$. A precursor of our method is an algorithm by Rónyai, which recovers the roots through Galois computations over number fields, relies on GRH, and comes with a polynomial bound of unspecified degree; in contrast, our online phase works entirely over $\\mathbb F_q$ and is unconditional. Moreover, $\\mathcal D$ can be certified deterministically and unconditionally in time polynomial in $\\log q$ and $Ξ(\\mathcal D)$. We conjecture that every $\\mathbb F_q$ of large enough characteristic admits a datum with $Ξ(\\mathcal D)\\leq(\\log q)^{O(1)}$, and support this with a heuristic and with experiments. Constructing such a datum in polynomial time remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32508",
  "author_total": 1,
  "authors": [
    {
      "name": "Pierre-Jean Spaenlehauer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/82wve69nf7m5d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32508",
  "fulltext_url": null,
  "id": "82wve69nf7m5d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32508",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.SC"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7128984615",
  "pdf_url": "https://arxiv.org/pdf/2610.32508",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/82wve69nf7m5d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32508",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32508"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Deterministic Root-Finding over Finite Fields with Elliptic Curves",
  "title_text": "Deterministic Root-Finding over Finite Fields with Elliptic Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/82wve69nf7m5d",
  "venue": "arXiv",
  "versions": []
}