{
  "abstract": "Polynomial modular reduction is central to binary finite-field arithmetic and repeated Frobenius powering. Top-down shift/XOR folding can use few operations when the nonleading support is sparse, but a tap near the leading term creates a long feedback chain. We formulate this recurrence as inversion of a nilpotent shift operator and factor its inverse by characteristic-two Frobenius powers. The resulting Frobenius-factorized reduction (FFR) applies to every monic binary modulus without materializing a reciprocal or dense reduction matrix, and its shifts can be generated online without a persistent modulus-specific schedule. For degree $m$ and nonempty nonleading support of size $s$, with nearest-tap distance $Δ_{\\min}$, FFR uses exactly $\\lceil\\log_2(m/Δ_{\\min})\\rceil$ sequential feedback stages and has scheduled work $O(ms(1+\\log(m/s)))$. A portable-C evaluation on 1,096 supports through degree $131072$ identifies distinct FFR, López--Dahab, and gf2x-backed Barrett regions. On four certified irreducible moduli, FFR makes complete Rabin irreducibility testing $1.35$--$8.04$ times faster than NTL and $1.60$--$6.81$ times faster than the matched Barrett implementation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08608",
  "author_total": 2,
  "authors": [
    {
      "name": "Junyu Zhou"
    },
    {
      "name": "Kaiyi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fypyhcm7gz611.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08608",
  "fulltext_url": null,
  "id": "fypyhcm7gz611",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08608",
  "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.SC"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1710|Cryptography and Residue Arithmetic"
    ]
  ],
  "openalex": "W7211991603",
  "pdf_url": "https://arxiv.org/pdf/2609.08608",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fypyhcm7gz611.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.08608",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08608"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Reduction Modulo Binary Polynomials with Logarithmic Feedback Depth",
  "title_text": "Reduction Modulo Binary Polynomials with Logarithmic Feedback Depth",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/fypyhcm7gz611",
  "venue": "arXiv",
  "versions": []
}