Definition
Access structure: [2]
Definition 1
Let \(\mathcal{P}\) be a set of parties. An access structure \({\Gamma }_{\mathcal{P}}\) is a subset of the powerset \({2}^{\mathcal{P}}\). (Each element \({\Gamma }_{\mathcal{P}}\) is considered trusted, e.g., has access to a shared secret.) \({\Gamma }_{\mathcal{P}}\) is monotone if for each element of \({\Gamma }_{\mathcal{P}}\) each superset belongs to \({\Gamma }_{\mathcal{P}}\), formally: when \(\mathcal{A}\subseteq \mathcal{B}\subseteq \mathcal{P}\) and \(\mathcal{A}\in {\Gamma }_{\mathcal{P}}\) then \(\mathcal{B}\in {\Gamma }_{\mathcal{P}}\).
Adversary structure: [1]
Definition 2
An adversary structure is the complement of an access structure, formally, if \({\Gamma }_{\mathcal{P}}\) is an access structure, then \({2}^{\mathcal{P}}\setminus {\Gamma }_{\mathcal{P}}\)is an adversary...
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Hirt M, Maurer U (2000) Player simulation and general adversary structures in perfect multiparty computation. J Cryptol 13(1):31–60
Ito M, Saito A, Nishizeki T (1987) Secret sharing schemes realizing general access structures. In: Proceedings of the IEEE global telecommunications conference, Globecom ’87, Tokyo. IEEE Communications Society, pp 99–102
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Desmedt, Y. (2011). Access Structure. 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_313
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