{
  "abstract": "Full-support local randomizers can incur unnecessary reporting distortion on ordered alphabets. We study randomized response whose output is restricted to a radius-$t$ neighborhood of the private input. Input-dependent supports create privacy leakage at their boundaries, so ordinary likelihood-ratio calibration is no longer sufficient. Under approximate local privacy restricted to input pairs at distance at most $H$, we derive the exact hockey-stick defect of the mechanism. We then solve the joint design of the truthful probability, off-truth probability, and support radius. The fixed-radius optimum is available in closed form, while the global optimum requires only a one-dimensional integer search. We characterize both the privacy floor and its inverse, namely the smallest defect attainable under a reporting-distortion budget. We further prove that stringent approximate privacy forces the support size to scale as $Ω(H/δ)$. Deterministic evaluations illustrate the resulting tradeoff.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15972",
  "author_total": 3,
  "authors": [
    {
      "name": "Amirreza Zamani"
    },
    {
      "name": "Parastoo Sadeghi"
    },
    {
      "name": "Mikael Skoglund"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9238t4dhwkdza.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15972",
  "fulltext_url": null,
  "id": "9238t4dhwkdza",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15972",
  "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|Privacy-Preserving Technologies in Data"
    ]
  ],
  "openalex": "W7221785698",
  "pdf_url": "https://arxiv.org/pdf/2610.15972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9238t4dhwkdza.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Exact Privacy--Distortion Design of Support-Constrained Randomized Response",
  "title_text": "Exact Privacy--Distortion Design of Support-Constrained Randomized Response",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/9238t4dhwkdza",
  "venue": "arXiv",
  "versions": []
}