{
  "abstract": "Direct evaluation of the defining product of the Cayley-Dickson algebra $A_n$ of rank $N:=2^n$ over a unital commutative ring $R$ in which $2$ is invertible requires a quadratic number of arithmetic operations. This paper gives a uniform bilinear algorithm for multiplication in this algebra costing $O(N \\log N)$ arithmetic operations in $R$ and $O(N)$ auxiliary storage when the doubling parameters of $A_n$ are units whose square roots lie in a fixed commutative coefficient algebra $S$ finitely generated as an $R$-module and admitting an $R$-linear map $π\\colon S \\to R$ with $π(1) = 1$. In particular, over $\\mathbb{R}$ and every finite field $\\mathbb{F}_{q}$ of odd characteristic, a quadratic field extension contains a square root of every possible doubling parameter, so the algorithm applies to every Cayley-Dickson tower over these fields, and the tensor rank of multiplication in $A_n$ over them is at most $(9n-21)2^n+28$ when $n\\ge2$. When the top doubling parameter is a non-square, as in the standard real tower, the bound is $(9n-15)2^{n-1}+10$. To the author's knowledge, the algorithm presented in this paper is the first uniform subquadratic algorithm for multiplication in $A_n$. When $R=\\mathbb{R}$ with every doubling parameter equal to $-1$, the real quasilinear schedule uses fewer total arithmetic operations than both direct multiplication and a uniform algorithm of Cariow-Cariowa for $n\\ge4$. An implementation for the standard real tower is available in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface. In single-core benchmarks against direct multiplication and the uniform Cariow-Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\\ge32$, for both single and batched products, and was over $17$ times faster than direct multiplication at $N=1024$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11588",
  "author_total": 1,
  "authors": [
    {
      "name": "Harrison Lemley"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/544vfz4bdf27t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11588",
  "fulltext_url": null,
  "id": "544vfz4bdf27t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11588",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "cs.SC"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7212303165",
  "pdf_url": "https://arxiv.org/pdf/2609.11588",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/544vfz4bdf27t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.11588",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11588"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Quasilinear multiplication in the Cayley-Dickson tower",
  "title_text": "Quasilinear multiplication in the Cayley-Dickson tower",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/544vfz4bdf27t",
  "venue": "arXiv",
  "versions": []
}