Abstract
We consider access structures over a set \(\mathcal {P}\) of n participants, defined by a parameter k with \(1 \le k \le n\) in the following way: a subset is authorized if it contains participants \(i,i+1,\ldots ,i+k-1\), for some \(i \in \{1,\ldots ,n-k+1\}\). We call such access structures, which may naturally appear in real applications involving distributed cryptography, (k, n)-consecutive.
We prove that these access structures are only ideal when \(k=1,n-1,n\). Actually, we obtain the same result that has been obtained for other families of access structures: being ideal is equivalent to being a vector space access structure and is equivalent to having an optimal information rate strictly bigger than \(\frac{2}{3}\). For the non-ideal cases, we give either the exact value of the optimal information rate, for \(k=n-2\) and \(k=n-3\), or some bounds on it.
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This work is partially supported by Spanish Ministry of Economy and Competitiveness, under Project MTM2016-77213-R.
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Herranz, J., Sáez, G. (2018). Secret Sharing Schemes for (k, n)-Consecutive Access Structures. In: Camenisch, J., Papadimitratos, P. (eds) Cryptology and Network Security. CANS 2018. Lecture Notes in Computer Science(), vol 11124. Springer, Cham. https://doi.org/10.1007/978-3-030-00434-7_23
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DOI: https://doi.org/10.1007/978-3-030-00434-7_23
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