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Shamir’s Threshold Scheme

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Encyclopedia of Cryptography and Security

Related Concepts

Secret Sharing Schemes; Threshold Cryptography

Definition

A technique by means of polynomials to let k people recover a secret, while any combination of up to k − 1 people does not even have partial information about that secret.

Background

In Ref. [1], A. Shamir proposed an elegant “polynomial” construction of a perfect threshold schemes . An (n, k)-threshold scheme is a particular case of a secret sharing scheme when any set of k or more participants can recover the secret exactly while any set of less than k particiants gains no additional, that is, a posteriori, information about the secret. Such threshold schemes are called perfect and they were constructed in Refs. [ 2,  1]. Shamir’s construction is the following.

Theory

Assume that the set S 0 of secrets is some finite field GF(q) of q elements (hence q should be prime power) and that the number of participants of SSS n < q. The dealer chooses n different nonzero elements (points) \({x}_{1},\ldots,{x}_{n} \in...

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Recommended Reading

  1. Shamir A (1979) How to share a secret. Commun ACM 22(1): 612–613

    Article  MATH  MathSciNet  Google Scholar 

  2. Blakley R (1979) Safeguarding cryptographic keys. In: Proceedings of AFIPS 1979 national computer conference, New York, vol 48, pp 313–317

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  3. McEliece RJ, Sarwate DV (1981) On secret sharing and Reed-Solomon codes. Commun ACM 24:583–584

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, 77843 - 3368, TX, USA

    G. R. Blakley (Dr.)

  2. Dobrushin Mathematical Lab, Institute for Information Transmission Problems RAS, B. Karetniy 19, 127994, Moscow, Russia

    Gregory Kabatiansky (Dr.)

Authors
  1. G. R. Blakley
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  2. Gregory Kabatiansky
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB, Eindhoven, The Netherlands

    Henk C. A. van Tilborg

  2. Center for Secure Information Systems, George Mason University, Fairfax, VA, 22030-4422, USA

    Sushil Jajodia

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© 2011 Springer Science+Business Media, LLC

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Blakley, G.R., Kabatiansky, G. (2011). Shamir’s Threshold Scheme. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_390

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