{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "karatsuba-ofman-multiplication-automata-1962",
    "title": "Karatsuba and Ofman: fewer pieces for large-number multiplication",
    "summary": "The1962 joint note presents two separately attributed binary-automaton multiplication constructions: Ofman’s grouped parallel/sequential tradeoff and Karatsuba’s recursive three-square construction, which breaks the quadratic-size pattern of school multiplication.\n\nSplitting a large number into halves seems to require four smaller products. The note reduces multiplication to squaring and combines three smaller squares instead, recursively reusing that saving. It also examines a different choice: how much hardware to allocate versus how much work to perform sequentially. The original treats machine size and elapsed machine steps separately and leaves the best joint tradeoff open.",
    "boundary": "Exact target is the 1962 two-page note’s binary-automaton constructions/proof schemes, not arbitrary unit-cost arithmetic or every modern big-integer library. Theorem 1 is explicitly Ofman’s grouped size/time construction; Theorem 2 is explicitly Karatsuba’s recursive squaring construction. Three smaller squares plus additions/shifts give the subquadratic-size mechanism; the carry footnote addresses the extra sum digit. N(machine size) and T(time) are different resources. Corrupted formula exponents prevent reliable transcription of every joint bound or timing exponent here; neither elementary separate lower bounds nor the constructions prove optimality or attain both best orders together. Original 294 distinguishes feedback-free representations from automata with reusable components, so no unconditional circuit tradeoff equivalence is inferred. Earlier Ofman addition/model definitions are cited but unread. Both Russian pages prose read; no independent full automaton reconstruction, clean-formula visual check or program/benchmark replay. The 1963 English translation is the same work, not another achievement.",
    "area": "Computer arithmetic",
    "topic": "Computing",
    "year": 1962,
    "lane": "HUMAN",
    "result": "ALGORITHM",
    "package_id": "karatsuba-ofman-multiplication-automata-1962",
    "target": {
      "record_id": "urn:uuid:fd16996e-b7c0-4587-ad92-320290b36db5",
      "version": 1,
      "content_hash": "sha256:03e75edd9a841bc3e2979de3812182a784cf4e2ef0ca5edbda75e72491292ea3"
    },
    "humans": [
      "Anatolii A. Karatsuba",
      "Yu. Ofman"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.",
    "agents": [],
    "related": [
      "strassen-gaussian-elimination-not-optimal-1969",
      "cooley-tukey-fft"
    ],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "wrs24-karatsuba-ofman-1962-original-mathnet",
        "title": "A.Karatsuba and Yu.Ofman, Умножение многозначных чисел на автоматах, Doklady145(2),293–294 (1962)",
        "url": "https://m.mathnet.ru/php/getFT.phtml?jrnid=dan&option_lang=eng&paperid=26729&what=fullt"
      },
      {
        "id": "wrs24-karatsuba-1962-author-bibliography",
        "title": "Steklov author bibliography:1962 Russian article and1963 English translation",
        "url": "https://homepage.mi-ras.ru/~karatsuba/list_e.html"
      }
    ],
    "points": "210000",
    "published_grade": {
      "id": "karatsuba-ofman-multiplication-automata-1962",
      "entry_id": "karatsuba-ofman-multiplication-automata-1962",
      "band": "A",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:fd16996e-b7c0-4587-ad92-320290b36db5",
        "version": 1,
        "content_hash": "sha256:03e75edd9a841bc3e2979de3812182a784cf4e2ef0ca5edbda75e72491292ea3"
      },
      "what_changed": "Introduces a three-recursive-square multiplication mechanism alongside separately attributed grouped machine-size/time tradeoffs.",
      "scope": "Introduces a three-recursive-square multiplication mechanism alongside separately attributed grouped machine-size/time tradeoffs. Exact target is the 1962 two-page note’s binary-automaton constructions/proof schemes, not arbitrary unit-cost arithmetic or every modern big-integer library. Theorem 1 is explicitly Ofman’s grouped size/time construction; Theorem 2 is explicitly Karatsuba’s recursive squaring construction. Three smaller squares plus additions/shifts give the subquadratic-size mechanism; the carry footnote addresses the extra sum digit. N(machine size) and T(time) are different resources. Corrupted formula exponents prevent reliable transcription of every joint bound or timing exponent here; neither elementary separate lower bounds nor the constructions prove optimality or attain both best orders together. Original 294 distinguishes feedback-free representations from automata with reusable components, so no unconditional circuit tradeoff equivalence is inferred. Earlier Ofman addition/model definitions are cited but unread. Both Russian pages prose read; no independent full automaton reconstruction, clean-formula visual check or program/benchmark replay. The 1963 English translation is the same work, not another achievement.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:fd16996e-b7c0-4587-ad92-320290b36db5",
        "version": 1,
        "content_hash": "sha256:03e75edd9a841bc3e2979de3812182a784cf4e2ef0ca5edbda75e72491292ea3"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-03T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "Anatolii A. Karatsuba",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "formulator"
          ],
          "source": {
            "citation": "A.Karatsuba and Yu.Ofman, Умножение многозначных чисел на автоматах, Doklady145(2),293–294 (1962)",
            "url": "https://m.mathnet.ru/php/getFT.phtml?jrnid=dan&option_lang=eng&paperid=26729&what=fullt"
          }
        },
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "Yu. Ofman",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "formulator"
          ],
          "source": {
            "citation": "A.Karatsuba and Yu.Ofman, Умножение многозначных чисел на автоматах, Doklady145(2),293–294 (1962)",
            "url": "https://m.mathnet.ru/php/getFT.phtml?jrnid=dan&option_lang=eng&paperid=26729&what=fullt"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "A.Karatsuba and Yu.Ofman, Умножение многозначных чисел на автоматах, Doklady145(2),293–294 (1962)",
            "url": "https://m.mathnet.ru/php/getFT.phtml?jrnid=dan&option_lang=eng&paperid=26729&what=fullt"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-1",
          "kind": "source_reference",
          "source": {
            "citation": "Steklov author bibliography:1962 Russian article and1963 English translation",
            "url": "https://homepage.mi-ras.ru/~karatsuba/list_e.html"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "Exact target is the 1962 two-page note’s binary-automaton constructions/proof schemes, not arbitrary unit-cost arithmetic or every modern big-integer library. Theorem 1 is explicitly Ofman’s grouped size/time construction; Theorem 2 is explicitly Karatsuba’s recursive squaring construction. Three smaller squares plus additions/shifts give the subquadratic-size mechanism; the carry footnote addresses the extra sum digit. N(machine size) and T(time) are different resources. Corrupted formula exponents prevent reliable transcription of every joint bound or timing exponent here; neither elementary separate lower bounds nor the constructions prove optimality or attain both best orders together. Original 294 distinguishes feedback-free representations from automata with reusable components, so no unconditional circuit tradeoff equivalence is inferred. Earlier Ofman addition/model definitions are cited but unread. Both Russian pages prose read; no independent full automaton reconstruction, clean-formula visual check or program/benchmark replay. The 1963 English translation is the same work, not another achievement.",
        "domain": "Computer arithmetic"
      },
      "sources": [
        {
          "citation": "A.Karatsuba and Yu.Ofman, Умножение многозначных чисел на автоматах, Doklady145(2),293–294 (1962)",
          "url": "https://m.mathnet.ru/php/getFT.phtml?jrnid=dan&option_lang=eng&paperid=26729&what=fullt"
        },
        {
          "citation": "Steklov author bibliography:1962 Russian article and1963 English translation",
          "url": "https://homepage.mi-ras.ru/~karatsuba/list_e.html"
        }
      ],
      "statement": "# Karatsuba and Ofman: fewer pieces for large-number multiplication\n\nThe1962 joint note presents two separately attributed binary-automaton multiplication constructions: Ofman’s grouped parallel/sequential tradeoff and Karatsuba’s recursive three-square construction, which breaks the quadratic-size pattern of school multiplication.\n\nSplitting a large number into halves seems to require four smaller products. The note reduces multiplication to squaring and combines three smaller squares instead, recursively reusing that saving. It also examines a different choice: how much hardware to allocate versus how much work to perform sequentially. The original treats machine size and elapsed machine steps separately and leaves the best joint tradeoff open.\n\n## Scientific scope\n\nExact target is the 1962 two-page note’s binary-automaton constructions/proof schemes, not arbitrary unit-cost arithmetic or every modern big-integer library. Theorem 1 is explicitly Ofman’s grouped size/time construction; Theorem 2 is explicitly Karatsuba’s recursive squaring construction. Three smaller squares plus additions/shifts give the subquadratic-size mechanism; the carry footnote addresses the extra sum digit. N(machine size) and T(time) are different resources. Corrupted formula exponents prevent reliable transcription of every joint bound or timing exponent here; neither elementary separate lower bounds nor the constructions prove optimality or attain both best orders together. Original 294 distinguishes feedback-free representations from automata with reusable components, so no unconditional circuit tradeoff equivalence is inferred. Earlier Ofman addition/model definitions are cited but unread. Both Russian pages prose read; no independent full automaton reconstruction, clean-formula visual check or program/benchmark replay. The 1963 English translation is the same work, not another achievement.\n\n## People and systems\n\nAnatolii A. Karatsuba; Yu. Ofman\n\n\n\n## Original sources\n\n- [A.Karatsuba and Yu.Ofman, Умножение многозначных чисел на автоматах, Doklady145(2),293–294 (1962)](https://m.mathnet.ru/php/getFT.phtml?jrnid=dan&option_lang=eng&paperid=26729&what=fullt)\n- [Steklov author bibliography:1962 Russian article and1963 English translation](https://homepage.mi-ras.ru/~karatsuba/list_e.html)\n",
      "summary": "The1962 joint note presents two separately attributed binary-automaton multiplication constructions: Ofman’s grouped parallel/sequential tradeoff and Karatsuba’s recursive three-square construction, which breaks the quadratic-size pattern of school multiplication.\n\nSplitting a large number into halves seems to require four smaller products. The note reduces multiplication to squaring and combines three smaller squares instead, recursively reusing that saving. It also examines a different choice: how much hardware to allocate versus how much work to perform sequentially. The original treats machine size and elapsed machine steps separately and leaves the best joint tradeoff open.",
      "title": "Karatsuba and Ofman: fewer pieces for large-number multiplication"
    },
    "record_id": "urn:uuid:fd16996e-b7c0-4587-ad92-320290b36db5",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:fd16996e-b7c0-4587-ad92-320290b36db5",
      "version": 1,
      "content_hash": "sha256:03e75edd9a841bc3e2979de3812182a784cf4e2ef0ca5edbda75e72491292ea3"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:3b250562e7bfe5d190b1a5b0aacfd18a369310a6e1cb6a685b2e3845289cb38a"
  },
  "target": {
    "record_id": "urn:uuid:fd16996e-b7c0-4587-ad92-320290b36db5",
    "version": 1,
    "content_hash": "sha256:03e75edd9a841bc3e2979de3812182a784cf4e2ef0ca5edbda75e72491292ea3"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "karatsuba-ofman-multiplication-automata-1962",
    "entry_id": "karatsuba-ofman-multiplication-automata-1962",
    "revision": 1,
    "title": "Karatsuba and Ofman: fewer pieces for large-number multiplication",
    "context": "Computer arithmetic",
    "track": "Computer arithmetic",
    "band": "A",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-04",
    "evidence_cutoff": "2026-10-04",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Introduces a three-recursive-square multiplication mechanism alongside separately attributed grouped machine-size/time tradeoffs.",
    "scope": "Introduces a three-recursive-square multiplication mechanism alongside separately attributed grouped machine-size/time tradeoffs. Exact target is the 1962 two-page note’s binary-automaton constructions/proof schemes, not arbitrary unit-cost arithmetic or every modern big-integer library. Theorem 1 is explicitly Ofman’s grouped size/time construction; Theorem 2 is explicitly Karatsuba’s recursive squaring construction. Three smaller squares plus additions/shifts give the subquadratic-size mechanism; the carry footnote addresses the extra sum digit. N(machine size) and T(time) are different resources. Corrupted formula exponents prevent reliable transcription of every joint bound or timing exponent here; neither elementary separate lower bounds nor the constructions prove optimality or attain both best orders together. Original 294 distinguishes feedback-free representations from automata with reusable components, so no unconditional circuit tradeoff equivalence is inferred. Earlier Ofman addition/model definitions are cited but unread. Both Russian pages prose read; no independent full automaton reconstruction, clean-formula visual check or program/benchmark replay. The 1963 English translation is the same work, not another achievement.",
    "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.",
    "references": [
      {
        "title": "A. Karatsuba and Yu. Ofman, Умножение многозначных чисел на автоматах, Doklady 145(2),293–294 (1962)",
        "url": "https://m.mathnet.ru/php/getFT.phtml?jrnid=dan&option_lang=eng&paperid=26729&what=fullt",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      },
      {
        "title": "Steklov author bibliography:1962 Russian article and 1963 English translation",
        "url": "https://homepage.mi-ras.ru/~karatsuba/list_e.html",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      }
    ],
    "roles": [
      {
        "who": "Anatolii A. Karatsuba",
        "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.",
        "credit": "SCIENCE"
      },
      {
        "who": "Yu. Ofman",
        "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.",
        "credit": "SCIENCE"
      }
    ],
    "target": {
      "record_id": "urn:uuid:fd16996e-b7c0-4587-ad92-320290b36db5",
      "version": 1,
      "content_hash": "sha256:03e75edd9a841bc3e2979de3812182a784cf4e2ef0ca5edbda75e72491292ea3"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "210000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": null,
  "reading_status": "SOURCE_LINKED_NOTE"
}