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    "title": "Shamir: recover a secret only when enough shares come together",
    "summary": "Uniform random polynomial coefficients over a finite field make any k valid shares sufficient to recover a secret, while fewer shares have secret-independent distributions.\n\nA vital key kept in one place can be lost; keeping complete copies in many places can expose it. Shamir gave a simple way to divide it so that a chosen minimum number of holders can recover it together. Shares are values of a random polynomial, and interpolation reconstructs its constant term. Below the threshold, the shares provide no extra information about that term under the stated random-generation assumptions. The construction is small and mathematically exact, but ordinary shares do not prove that a holder is honest or that a submitted share is correct. Secure distribution, valid identifiers and proper randomness remain necessary. The paper credits a different contemporary threshold scheme by Blakley. This account does not independently audit a deployed secret-sharing product.",
    "boundary": "The secret is encoded in a finite field, with distinct nonzero participant indices and independently uniform polynomial coefficients. Any k consistent shares reconstruct it; fewer than k reveal no information under that randomization model. The original prime-field construction requires a modulus large enough for the secret and participant indices. Reconstruction assumes honest, intact shares. Detecting a malicious dealer, authenticating shares and verifiable secret sharing are later/separate mechanisms. Blakley developed a different threshold construction independently. WRS has not performed an implementation/security audit.",
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    "year": 1979,
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        "title": "Adi Shamir: How to Share a Secret (CACM1979 original)",
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        "description": "The secret is encoded in a finite field, with distinct nonzero participant indices and independently uniform polynomial coefficients. Any k consistent shares reconstruct it; fewer than k reveal no information under that randomization model. The original prime-field construction requires a modulus large enough for the secret and participant indices. Reconstruction assumes honest, intact shares. Detecting a malicious dealer, authenticating shares and verifiable secret sharing are later/separate mechanisms. Blakley developed a different threshold construction independently. WRS has not performed an implementation/security audit.",
        "domain": "Computer science"
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      "sources": [
        {
          "citation": "Adi Shamir: How to Share a Secret (CACM1979 original)",
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      "statement": "# Shamir: recover a secret only when enough shares come together\n\nUniform random polynomial coefficients over a finite field make any k valid shares sufficient to recover a secret, while fewer shares have secret-independent distributions.\n\nA vital key kept in one place can be lost; keeping complete copies in many places can expose it. Shamir gave a simple way to divide it so that a chosen minimum number of holders can recover it together. Shares are values of a random polynomial, and interpolation reconstructs its constant term. Below the threshold, the shares provide no extra information about that term under the stated random-generation assumptions. The construction is small and mathematically exact, but ordinary shares do not prove that a holder is honest or that a submitted share is correct. Secure distribution, valid identifiers and proper randomness remain necessary. The paper credits a different contemporary threshold scheme by Blakley. This account does not independently audit a deployed secret-sharing product.\n\n## Scientific scope\n\nThe secret is encoded in a finite field, with distinct nonzero participant indices and independently uniform polynomial coefficients. Any k consistent shares reconstruct it; fewer than k reveal no information under that randomization model. The original prime-field construction requires a modulus large enough for the secret and participant indices. Reconstruction assumes honest, intact shares. Detecting a malicious dealer, authenticating shares and verifiable secret sharing are later/separate mechanisms. Blakley developed a different threshold construction independently. WRS has not performed an implementation/security audit.\n\n## People and systems\n\nAdi Shamir\n\n\n\n## Original sources\n\n- [Adi Shamir: How to Share a Secret (CACM1979 original)](https://people.cs.pitt.edu/~litman/courses/cs2001/lectures/lecture-02-p612-shamir.pdf)\n",
      "summary": "Uniform random polynomial coefficients over a finite field make any k valid shares sufficient to recover a secret, while fewer shares have secret-independent distributions.\n\nA vital key kept in one place can be lost; keeping complete copies in many places can expose it. Shamir gave a simple way to divide it so that a chosen minimum number of holders can recover it together. Shares are values of a random polynomial, and interpolation reconstructs its constant term. Below the threshold, the shares provide no extra information about that term under the stated random-generation assumptions. The construction is small and mathematically exact, but ordinary shares do not prove that a holder is honest or that a submitted share is correct. Secure distribution, valid identifiers and proper randomness remain necessary. The paper credits a different contemporary threshold scheme by Blakley. This account does not independently audit a deployed secret-sharing product.",
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    "id": "shamir-polynomial-threshold-secret-sharing-1979",
    "entry_id": "shamir-polynomial-threshold-secret-sharing-1979",
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    "title": "Shamir: recover a secret only when enough shares come together",
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    "what_changed": "Uniform random polynomial coefficients over a finite field make any k valid shares sufficient to recover a secret, while fewer shares have secret-independent distributions.",
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    "references": [
      {
        "title": "Adi Shamir: How to Share a Secret (CACM 1979 original)",
        "url": "https://people.cs.pitt.edu/~litman/courses/cs2001/lectures/lecture-02-p612-shamir.pdf",
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