What is recorded here?
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. A 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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Scientific scope
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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People and systems
Adi Shamir
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English catalogue narrative
Shamir: recover a secret only when enough shares come together
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.
A 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.
Scientific scope
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.
People and systems
Adi Shamir