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Average complexities of access structures on five participants

• In this paper, we consider the 12 access structures on five participants for which determining the exact values of the average complexities remained as open problems in Jackson and Martin's paper [6]. We establish the exact values of the average complexities of these access structures.
Mathematics Subject Classification: 94A62, 94A17.

 Citation:

•  [1] C. Blundo, A. De Santis, D. R. Stinson and U. Vaccaro, Graph decompositions and secret sharing schemes, J. Cryptol., 8 (1995), 39-64.doi: 10.1007/BF00204801. [2] R. M. Capocelli, A. De Santis, L. Gargano and U. Vaccaro, On the size of shares of secret sharing schemes, J. Cryptol., 6 (1993), 157-168. [3] T. M. Cover and J. A. Thomas, "Elements of Information Theory,'' $2^{nd}$ edition, Wiley, New York, 2006. [4] L. Csirmaz, The size of a share must be large, J. Cryptol., 10 (1997), 223-231.doi: 10.1007/s001459900029. [5] W.-A. Jackson and K. M. Martin, Geometric secret sharing schemes and their duals, Des. Codes Cryptogr., 4 (1994), 83-95.doi: 10.1007/BF01388562. [6] W.-A. Jackson and K. M. Martin, Perfect secret sharing schemes on five participants, Des. Codes Cryptogr., 9 (1996), 267-286.doi: 10.1007/BF00129769. [7] J. Martí-Farré and C. Padró, On Secret Sharing Schemes, Matroids and Polymatroids, J. Math. Cryptol., 4 (2010), 95-120.doi: 10.1515/JMC.2010.004. [8] C. Padró and G. Sáez, Lower bounds on the information rate of secret sharing schemes with homogeneous access structure, Inform. Process. Lett., 83 (2002), 345-351.doi: 10.1016/S0020-0190(02)00213-2. [9] C. Padró and L. Vázquez, Finding lower bounds on the complexity of secret sharing schemes by linear programming, in "9th Latin American theoretical informatics symposium", (2010), 344-355.doi: 10.1007/978-3-642-12200-2_31.

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