{
  "abstract": "We consider private retrieval of the matrix product $A^\\top B^{(θ)}$ from replicated $P_c$-colluding servers when $B^{(\\ell)}=A+N^{(\\ell)}$, with the innovations mutually independent and independent of the user-held $A\\in\\mathbb{F}_q^{m\\times m}$. The servers store the full library, correlated with $A$, which changes both the communication problem and the security definition: a secrecy condition including the stored library cannot hold, as the library itself partly reveals $A$. We therefore measure only additional leakage and require index privacy conditioned on $A$; both reduce to classical notions under independence. For $A$ of rank $k<m$ we prove, with no structural hypothesis on the map, $D^*\\geq H(A^\\top B^{(θ)}\\mid A)/C_{\\mathrm{PIR}}(N,P_c,L)-o(km)$, where $C_{\\mathrm{PIR}}(N,P_c,L)$ is the replicated $P_c$-colluding private-information-retrieval capacity. The proof rests on a colluding converse for servers storing arbitrary refinements of the demanded messages, whose answers need not be measurable functions of the functional-message library, and a privacy-transfer characterization. Publishing a canonical row-space representative makes the bound achievable for arbitrary independent and identically distributed innovations under a public-setup measure. Under a strengthened observational model (exact correctness, common conditional library support, and full-transcript security), concealment forces $D\\geq H(G_m N^{(θ)})/C_{\\mathrm{PIR}}$, where $G_m$ spans the linear join of all row spaces in the support of the operand distribution; this extends to vanishing error under a simultaneous-decodability assumption. In the full-rank case product recovery reduces to innovation recovery: $D^*=H(N^{(θ)})/C_{\\mathrm{PIR}}+o(m^2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07325",
  "author_total": 2,
  "authors": [
    {
      "name": "Malihe Aliasgari"
    },
    {
      "name": "Yousef Nejatbakhsh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6r9xg3ywdrsmp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07325",
  "fulltext_url": null,
  "id": "6r9xg3ywdrsmp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07325",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07325",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6r9xg3ywdrsmp.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07325",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07325"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Functional Converses for Private Matrix-Product Retrieval under Library-Wide Decoder-Side Correlation",
  "title_text": "Functional Converses for Private Matrix-Product Retrieval under Library-Wide Decoder-Side Correlation",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/6r9xg3ywdrsmp",
  "venue": "arXiv",
  "versions": []
}