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    "summary": "Rabin gives a public-key function based on modular quadratic equations and a randomized reduction showing that inversion on a substantial fraction of its invertible unit-domain outputs enables factorization of the modulus.\n\nA public-key function should be easy to evaluate and hard to reverse without a private key. Michael O. Rabin studies a quadratic function modulo a product of two distinct odd primes. Knowing the two factors makes inversion possible using modular roots and the Chinese remainder theorem. Conversely, an algorithm that finds a root on enough suitable outputs can be used in a randomized procedure to recover a factor. This makes the mathematical relationship to factorization explicit rather than merely assuming that inversion requires factoring. The report also describes randomized message suffixes and hashing for signatures. Those protocol steps require their own assumptions: the primitive reduction does not prove modern security for every hash function, message distribution or adversarial interaction. Nor does the report prove that factorization is inherently hard. The contribution is represented through the 1979 report, with its reference to Rabin’s earlier 1978 signature work retained rather than claiming that every part was first invented in 1979.",
    "boundary": "The inversion/factorization equivalence belongs to the stated modular-square trapdoor construction. Extracting roots has ambiguities and protocol encodings need additional conditions. This is not unrestricted chosen-ciphertext security, authenticated encryption or a current deployment recommendation. Later padding/signature schemes and earlier factorization methods retain separate credit.",
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    "year": 1979,
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        "title": "Rabin1979MIT-LCS-TR212: Digitalized signatures and public-key functions as intractable as factorization",
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        "description": "The inversion/factorization equivalence belongs to the stated modular-square trapdoor construction. Extracting roots has ambiguities and protocol encodings need additional conditions. This is not unrestricted chosen-ciphertext security, authenticated encryption or a current deployment recommendation. Later padding/signature schemes and earlier factorization methods retain separate credit.",
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      "sources": [
        {
          "citation": "Rabin1979MIT-LCS-TR212: Digitalized signatures and public-key functions as intractable as factorization",
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      "statement": "# Rabin: relating trapdoor-function inversion to factorization\n\nRabin gives a public-key function based on modular quadratic equations and a randomized reduction showing that inversion on a substantial fraction of its invertible unit-domain outputs enables factorization of the modulus.\n\nA public-key function should be easy to evaluate and hard to reverse without a private key. Michael O. Rabin studies a quadratic function modulo a product of two distinct odd primes. Knowing the two factors makes inversion possible using modular roots and the Chinese remainder theorem. Conversely, an algorithm that finds a root on enough suitable outputs can be used in a randomized procedure to recover a factor. This makes the mathematical relationship to factorization explicit rather than merely assuming that inversion requires factoring. The report also describes randomized message suffixes and hashing for signatures. Those protocol steps require their own assumptions: the primitive reduction does not prove modern security for every hash function, message distribution or adversarial interaction. Nor does the report prove that factorization is inherently hard. The contribution is represented through the 1979 report, with its reference to Rabin’s earlier 1978 signature work retained rather than claiming that every part was first invented in 1979.\n\n## Scientific scope\n\nThe inversion/factorization equivalence belongs to the stated modular-square trapdoor construction. Extracting roots has ambiguities and protocol encodings need additional conditions. This is not unrestricted chosen-ciphertext security, authenticated encryption or a current deployment recommendation. Later padding/signature schemes and earlier factorization methods retain separate credit.\n\n## People and systems\n\nMichael O. Rabin\n\n\n\n## Original sources\n\n- [Rabin1979MIT-LCS-TR212: Digitalized signatures and public-key functions as intractable as factorization](https://publications.csail.mit.edu/lcs/pubs/pdf/MIT-LCS-TR-212.pdf)\n- [Bellare/Rogaway1996: exact security of digital signatures, author abstract](https://www.cs.ucdavis.edu/~rogaway/papers/exact.html)\n",
      "summary": "Rabin gives a public-key function based on modular quadratic equations and a randomized reduction showing that inversion on a substantial fraction of its invertible unit-domain outputs enables factorization of the modulus.\n\nA public-key function should be easy to evaluate and hard to reverse without a private key. Michael O. Rabin studies a quadratic function modulo a product of two distinct odd primes. Knowing the two factors makes inversion possible using modular roots and the Chinese remainder theorem. Conversely, an algorithm that finds a root on enough suitable outputs can be used in a randomized procedure to recover a factor. This makes the mathematical relationship to factorization explicit rather than merely assuming that inversion requires factoring. The report also describes randomized message suffixes and hashing for signatures. Those protocol steps require their own assumptions: the primitive reduction does not prove modern security for every hash function, message distribution or adversarial interaction. Nor does the report prove that factorization is inherently hard. The contribution is represented through the 1979 report, with its reference to Rabin’s earlier 1978 signature work retained rather than claiming that every part was first invented in 1979.",
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    "id": "rabin-trapdoor-function-inversion-factorization-1979",
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    "what_changed": "Rabin gives a public-key function based on modular quadratic equations and a randomized reduction showing that inversion on a substantial fraction of its invertible unit-domain outputs enables factorization of the modulus.",
    "scope": "Rabin gives a public-key function based on modular quadratic equations and a randomized reduction showing that inversion on a substantial fraction of its invertible unit-domain outputs enables factorization of the modulus. The inversion/factorization equivalence belongs to the stated modular-square trapdoor construction. Extracting roots has ambiguities and protocol encodings need additional conditions. This is not unrestricted chosen-ciphertext security, authenticated encryption or a current deployment recommendation. Later padding/signature schemes and earlier factorization methods retain separate credit.",
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        "title": "Rabin 1979 MIT-LCS-TR 212: Digitalized signatures and public-key functions as intractable as factorization",
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