Abstract
A shared secret scheme is normally specified in terms of a desired security, Pd, and a concurrence scheme, Γ. The concurrence scheme (aka access structure) identifies subsets of participants (also called trustees or shareholders) each of which should be able to cooperatively recover the secret and/or initiate the controlled action. The security requirement is expressed as the maximum acceptable probability, Pd, that the secret can be exposed or the controlled action initiated by a collection of persons that doesn’t include at least one of the authorized subsets identified in the concurrence scheme. A concurrence scheme is said to be monotone if every set of participants that includes one or more sets from Γ is also able to recover the secret. The closure of Γ, denoted by \( \hat \Gamma \) is the collection of all supersets (not necessarily proper) of the sets in Γ, i.e., the collection of all sets of participants that can recover the secret and/or initiate the controlled action. A shared secret scheme implementing a concurrence scheme Γ is said to be perfect if the probability of recovering the secret is the same for every set, C, of participants: C \( \hat \Gamma \). Since, in particular, C could consist of only nonparticipants, i.e., of persons with no insider information about the secret, the probability, P, of an unauthorized recovery of the secret in a perfect scheme is just the probability of being able to “guess” the secret using only public information about Γ and the shared secret scheme implementing Γ: P ≤ Ptd.
This work performed at Sandia National Laboratories supported by the U. S. Department of Energy under contract no. DE-AC04-76DP00789.
Chapter PDF
Similar content being viewed by others
References
J. Benaloh and J. Leichter, “Generalized Secret Sharing and Monotone Functions,” Crypto’88, Santa Barbara, CA, August 21–25, 1988, Advances in Cryptology, Ed. by G. Goos and J. Hartmanis, Vol. 403, Springer-Verlag, Berlin, 1990, pp. 27–35.
E. F. Brickell and D. R. Stinson, “The Detection of Cheaters in Threshold Schemes,” Crypto’88, Santa Barbara, CA, August 21–25, 1988, Advances in Cryptology, Ed. by G. Goos and J. Hartmanis, Vol. 403, Springer-Verlag, Berlin, 1990, pp. 564–577.
B. Chor, S. Goldwasser, S. Micali and B. Awerbuch, “Verifiable Secret Sharing and Achieving Simultaneity in the Presence of Faults,” Proc. 26th IEEE Symp. Found. Comp. Sci., Portland, OR, October 1985, pp. 383–395.
I. Ingemarsson and G. J. Simmons, “How Mutually Distrustful Parties Can Set Up a Mutually Trusted Shared Secret Scheme,” International Association for Cryptologic Research (IACR) Newsletter, Vol. 7, No. 1, January 1990, pp. 4–7.
I. Ingemarsson and G. J. Simmons, “A Protocol to Set Up Shared Secret Schemes Without the Assistance of a Mutually Trusted Party,” to be presented at Eurocrypt’90, Aarhus, Denmark, May 21–24, 1990, Advances in Cryptology, to appear.
M. Ito, A. Saito and T. Nishizeki, “Secret Sharing Scheme Realizing General Access Structure,” (in English) Proc. IEEE Global Telecommunications Conf., Globecom’87, Tokyo, Japan, 1987, IEEE Communications Soc. Press, Washington, D.C., 1987, pp. 99–102. Also to appear in Trans. IEICE Japan, Vol. J71-A, No. 8, 1988 (in Japanese).
G. J. Simmons, “Robust Shared Secret Schemes or ‘How to be Sure You Have the Right Answer Even Though You Don’t Know the Question’,” 18th Annual Conference on Numerical Mathematics and Computing, Sept. 29–Oct. 1, 1988, Winnipeg, Manitoba, Canada, Congressus Numerantium, Vol. 68, May 1989, pp. 215–248.
G. J. Simmons, “Prepositioned Shared Secret and/or Shared Control Schemes,” Eurocrypt’89, Houthalen, Belgium, April 11–13, 1989, Advances in Cryptology, to appear.
M. Tompa and H. Woll, “How to Share a Secret with Cheaters,” Crypto’86, Santa Barbara, CA, Aug. 19–21, 1986, Advances in Cryptology, Vol. 263, Ed. by A. M. Odlyzko, Springer-Verlag, Berlin, 1986, pp. 261–265; also Journal of Cryptology, Vol. 1, No. 2, 1988, pp. 133–138.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Simmons, G.J. (1991). Geometric Shared Secret and/or Shared Control Schemes. In: Menezes, A.J., Vanstone, S.A. (eds) Advances in Cryptology-CRYPTO’ 90. CRYPTO 1990. Lecture Notes in Computer Science, vol 537. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-38424-3_16
Download citation
DOI: https://doi.org/10.1007/3-540-38424-3_16
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54508-8
Online ISBN: 978-3-540-38424-3
eBook Packages: Springer Book Archive