{
  "abstract": "We introduce function tables, an entrywise representation of the coefficient functions that appear in the worker responses of a secure distributed matrix multiplication (SDMM) scheme. We work under the outer-product partition, with $K$ row blocks, $L$ column blocks, and privacy against any $T$ colluding workers, in the general model of linear encoding and linear decoding. In this representation, privacy is a rank condition on the data and mask coefficients, and decodability is linear independence of the desired entries modulo the nuisance space. Degree tables, cyclic-addition tables, and algebraic-geometry constructions are the special cases obtained by restricting the coefficient functions to a structured family; we impose no such restriction, so our converses bind every linear scheme. For $T=1$, we determine the exact optimum over every finite field $\\mathbb{F}_q$: it is $KL+K+L$ when $q\\geq3$, and $KL+K+L+1$ over $\\mathbb{F}_2$, where the identity $z^2=z$ forces one more worker. For arbitrary $T$, we prove $N\\geq KL+K+L$ and $N\\geq\\max\\{K,L\\}+T$ with no MDS hypothesis on the masks; the first is stronger than the previously known bound $KL+\\max\\{K,L\\}+2T-1$ whenever $\\min\\{K,L\\}\\geq2T$. We then reduce field feasibility exactly to MDS existence: a scheme exists over $\\mathbb{F}_q$ if and only if an $[\\max\\{K,L\\}+T,T]$ linear MDS code does, and whenever it does, a Cartesian construction attains $N=(K+T)(L+T)$ over that same field. For $T=2$ this makes $q\\geq\\max\\{K,L\\}+1$ necessary and sufficient, and we give a projective-line construction with $N=KL+K+L+2$ whenever $KL+K+L$ divides $q-1$; for $K,L\\geq2$ it matches the best known worker count while requiring only an element of order $KL+K+L$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08154",
  "author_total": 6,
  "authors": [
    {
      "name": "Rafael G. L. D'Oliveira"
    },
    {
      "name": "Giulia Gaggero"
    },
    {
      "name": "Arturo Jaramillo Gil"
    },
    {
      "name": "Hiram H. López"
    },
    {
      "name": "Cecilia Martínez-Reyes"
    },
    {
      "name": "Divyesh Vaghasiya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cb68yw9a5c1z6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08154",
  "fulltext_url": null,
  "id": "cb68yw9a5c1z6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08154",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Cryptography and Data Security"
    ]
  ],
  "openalex": "W7212073298",
  "pdf_url": "https://arxiv.org/pdf/2609.08154",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cb68yw9a5c1z6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08154",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08154"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Function Tables for Secure Distributed Matrix Multiplication",
  "title_text": "Function Tables for Secure Distributed Matrix Multiplication",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/cb68yw9a5c1z6",
  "venue": "arXiv",
  "versions": []
}