{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R16",
  "content_hash": null,
  "slug": "apn256-claim-open-status",
  "type": "claim",
  "title": "Existence of an APN permutation on F_256 remains open",
  "summary": "No accepted construction or unrestricted nonexistence proof was found in the primary literature through 2026-07-25.",
  "relevance": "For An APN permutation of the 256-element field, record apn256-claim-open-status (“Existence of an APN permutation on F_256 remains open”) records a bound, answer, status fact, or structural consequence. The record states: No accepted construction or unrestricted nonexistence proof was found in the primary literature through 2026-07-25.",
  "relevance_source": "recorded",
  "body": "The answer remains open. A witness must be a bijection \\(F:\\mathbb F_{2^8}\\to\\mathbb F_{2^8}\\) satisfying\n\\[\n\\max_{a\\ne0,\\,b}\\#\\{x:F(x+a)+F(x)=b\\}=2.\n\\]\nThe complete search space contains \\(256!\\) lookup tables, so every published computation uses structural restrictions or selected CCZ-equivalence classes.\n\nBeierle, Brinkmann, and Leander searched APN permutations admitting a nontrivial linear self-equivalence. Their dimension-eight search exhausted all except a few self-equivalence classes and found no witness. Beierle, Langevin, Leander, Polujan, and Rasoolzadeh later generated 3,775,599 inequivalent quadratic APN functions in dimension eight. None of those functions is CCZ-equivalent to a permutation.\n\nA June 2026 preprint by Kuznetsov found four further quadratic APN CCZ-classes absent from the 2025 database. Its displayed functions are nonpermutations, and the paper leaves open whether any of the four CCZ-classes contains a permutation. This latest result preserves the unrestricted open status.",
  "status": "open",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "existence of an APN permutation on the 256-element binary field, equivalently an 8-bit APN permutation",
    "bounds": {
      "dimension": {
        "min": 8,
        "max": 8
      },
      "field_size": {
        "min": 256,
        "max": 256
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2606.11967",
      "locator": "Oleksandr Kuznetsov, Quadratic APN Functions in Dimension 8 via Gröbner Basis Search in a Self-Equivalence Subspace, arXiv:2606.11967v1 (2026), Sections VII-B, VIII, and IX; status cross-checked against the 2021 and 2025 primary computations listed in metadata"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2606.11967",
    "locator": "Oleksandr Kuznetsov, Quadratic APN Functions in Dimension 8 via Gröbner Basis Search in a Self-Equivalence Subspace, arXiv:2606.11967v1 (2026), Sections VII-B, VIII, and IX; status cross-checked against the 2021 and 2025 primary computations listed in metadata"
  },
  "models": [],
  "relations": [
    {
      "slug": "R15",
      "title": "A hypothetical witness has tightly restricted Boolean components",
      "object_type": "claim",
      "relation": "constrains",
      "direction": "incoming"
    },
    {
      "slug": "R13",
      "title": "Exact differential-uniformity verifier and power-permutation scan",
      "object_type": "artifact",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R14",
      "title": "Classification and computational-search audit",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "apn-permutation-f256",
      "title": "apn permutation f256",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}