{
  "abstract": "We determine the information complexity of matrix multiplication over finite fields under independent uniform inputs, up to a universal additive constant. The binary inner-product case resolves the open question of Braverman, Garg, Pankratov, and Weinstein on the optimal dependence of information complexity on the error probability. For inputs in $\\mathbb F_2^m$ and error at most $\\varepsilon\\in[0,1/2]$, the information complexity with output at Bob lies between $(1-2\\varepsilon)m-c_0$ and $(1-2\\varepsilon)m$, where $c_0<4.772$ is a universal constant. For independent uniform matrices $X\\in\\mathbb F_q^{t\\times m}$ and $Y\\in\\mathbb F_q^{s\\times m}$, where $t\\le s$, computing $XY^{\\mathsf T}$ with output at Bob and error at most $\\varepsilon\\in[0,1]$ has information complexity $(1-q^{ts}\\varepsilon/(q^{ts}-1))^+mt\\log_2q$, up to an additive gap of less than $9.544$ bits. This gap is independent of the matrix dimensions, field size, and error. At zero error, the information complexity is exactly $mt\\log_2q$. The information lower bounds for both inner product and matrix multiplication follow from a new Shannon-to-min-entropy theorem for convolution on discrete groups. Let $Q_1,\\ldots,Q_k$ be the finitely supported distributions of independent random variables taking values in a discrete group, and let $s$ be the largest support size. We show that if $k\\ge\\lceil\\lfloor\\log_2s\\rfloor\\ln2\\rceil$, then the min-entropy of their ordered product is at least the average Shannon entropy of the $Q_i$ minus $c_0$ bits. This holds for arbitrary discrete groups, including infinite and noncommutative groups, even when the input distributions differ. The leading coefficient of the logarithmic threshold is optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02684",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaohong Cai"
    },
    {
      "name": "Mingyang Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3kxg4tqs3zxf1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02684",
  "fulltext_url": null,
  "id": "3kxg4tqs3zxf1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02684",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02684",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3kxg4tqs3zxf1.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02684",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02684"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Information Complexity of Inner Product and Matrix Multiplication via Shannon-to-Min-Entropy Conversion",
  "title_text": "Information Complexity of Inner Product and Matrix Multiplication via Shannon-to-Min-Entropy Conversion",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/3kxg4tqs3zxf1",
  "venue": "arXiv",
  "versions": []
}