Abstract
Given a set of participants we wish to distribute information relating to a secret in such a way that only specified groups of participants can reconstruct the secret. We consider here a special class of such schemes that can be described in terms of finite geometries as first proposed by Simmons. We formalize the Simmons model and show that given a geometric scheme for a particular access structure it is possible to find another geometric scheme whose access structure is the dual of the original scheme, and which has the same average and worst-case information rates as the original scheme. In particular this shows that if an ideal geometric scheme exists then an ideal geometric scheme exists for the dual access structure.
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References
Benaloh, J. and Leichter, J. 1990. Generalized secret sharing and monotone functions. Advances in cryptology—Crypto '88, Lecture Notes inComput. Sci. 403:27–35.
Beutelspacher, A. and Vedder, K. 1989. Geometric structures as threshold schemes.Proc. of 1986 IMA Conf. on Cryptography and Coding, Oxford, Clarendon Press pp. 255–268.
Blakley, G.R. 1979. Safeguarding cryptographic keys.Proceedings of AFIPS 1979 National Computer Conference 48:313–317.
Blundo, C. 1991. Secret sharing schemes for access structures based on graphs. Tesi di Laurea, University of Salerno, Italy.
Brickell, E.F. and Davenport, D.M. 1991. On the classification of ideal secret sharing schemes.J. Cryptology 2:123–124.
Brickell, E.F. and Stinson, D.R. 1992. Some improved bounds on the information rate of perfect secret sharing schemes.J. Cryptology 2:153–166.
Hirschfeld, J.W.P. 1979.Projective Geometries over Finite Fields. Oxford, Clarendon Press.
Ito, M. Saito, A., and Nishizedi, T. 1987. Secret sharing scheme realizing general access structure.Proceedings IEEE Global Telecom. Conf., Globecom '87, Washington, D.C.: IEEE Comm. Soc. Press 99–102.
Jackson, W.-A. and Martin, K.M. On ideal secret sharing schemes. Preprint.
Martin, K.M. New Secret Sharing Schemes from Old. To appear inJ. Combin. Math. Combin. Comput.
Shamir, A. 1979. How to share a secret.Comm. ACM 22 11:612–613.
Simmons, G.J. 1990. How to (really) share a secret. Advances in Cryptology—Crypto '88, Lectures Notes inComput. Sci. 403:390–448.
Simmons, G.J., Jackson, W.-A., and Martin, K. 1991. The Geometry of Shared Secret Schemes. Bull. Inst. Combin. Appl. 1:71–88.
Simmons, G.J. 1992. An introduction to shared secret and/or shared control schemes and their applications.Contemporary Cryptology: The Science of Information Integrity, Piscataway, N.J.: IEEE Press.
Stinson, D.R. 1992. An explication of secret sharing schemes.Des. Codes Crptogr. 2:357–390.
Uehara, T., Nishizeki, T., Okamoto, E., and Nakamura, K. 1986. A secret sharing system with matroidal access structure.Trans. IECE Japan J69-A 9:1124–1132.
Welsh, D.J.A. 1976.Matroid Theory. London: Academic Press Inc. Ltd.
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Communicated by D. Jungnickel
This work was supported by the Science and Engineering Research Council Grant GR/G 03359.
This work was supported by the Australian Research Council.
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Jackson, WA., Martin, K.M. Geometric secret sharing schemes and their duals. Des Codes Crypt 4, 83–95 (1994). https://doi.org/10.1007/BF01388562
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DOI: https://doi.org/10.1007/BF01388562