{
  "abstract": "Let $N=pq$ be a semiprime and let $\\ell\\nmid N$ be an odd prime. For $a\\ne0$ modulo $\\ell$, put $H_a(N;\\ell)=\\{ax+Nx^{-1}:x\\in\\F_\\ell^*\\}$. For a primitive root $γ$, the size of $H_γ(N;\\ell)\\cup H_{γ^2}(N;\\ell)$ is determined by the Frobenius trace of $Y^2=(X^2-1)(X^2-γ)$. We study the resulting trace-selection problem through finite certificates, a congruence modulo $8$, and a GRH-conditional positive-density family from the CM specialization $γ=2$. Independently of this conditional result, we obtain a deterministic factorization algorithm from the residue-known coefficient lattice of Harvey and Hittmeir (\\emph{Math. Comp.}, 2022). The actual babystep index is a nondegenerate quadratic image modulo each selected prime divisor of the auxiliary modulus. After a computable translation, every such image avoids a fixed set of $(\\ell-χ(-1))/4$ residues. This forbidden set is optimal among translated sets that must handle both square classes of the leading coefficient. Gauss reduction, a nearby lattice-vector selection, arbitrary-point multipoint evaluation, and the unrestricted large-order algorithm of Harvey and Hittmeir (arXiv, 2026) give the bit-complexity bound $N^{1/5}\\exp(-(\\log(4/3)/50+o(1))\\log N/\\log\\log N)$. Its secondary factor is asymptotically smaller than every fixed negative power of $\\log N$, improving the general bound of Harvey and Hittmeir (\\emph{Math. Comp.}, 2022) and the balanced-semiprime bound of Gao et al.\\ (arXiv, 2025).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06580",
  "author_total": 1,
  "authors": [
    {
      "name": "Pantelimon Stănică"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bpeh75t625n2j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06580",
  "fulltext_url": null,
  "id": "bpeh75t625n2j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06580",
  "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",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06580",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bpeh75t625n2j.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06580",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06580"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Improved deterministic integer factorization by quadratic-image sieving",
  "title_text": "Improved deterministic integer factorization by quadratic-image sieving",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/bpeh75t625n2j",
  "venue": "arXiv",
  "versions": []
}