Optimal Information Rate of Secret Sharing Schemes on Trees

Csirmaz, László; Tardos, Gábor

doi:10.1109/tit.2012.2236958

Cited by 21 publications

(17 citation statements)

References 17 publications

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“…Due to the difficulty of finding general bounds, a number of works have considered this problem for particular families of perfect access structures. Recent examples are [3,23,28]. We determined the optimal information ratio for a family of non-perfect access functions, the uniform ones.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

On the Information Ratio of Non-perfect Secret Sharing Schemes

et al. 2016

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A secret sharing scheme is non-perfect if some subsets of players that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes and the construction of efficient linear non-perfect secret sharing schemes. To this end, we extend the known connections between matroids, polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information on the secret value that is obtained by each subset of players. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, access functions whose values depend only on the number of players, generalize the threshold access structures. The optimal information ratio of the uniform access functions with rational values has been determined by Yoshida, Fujiwara and Fossorier. By using the tools that are described in our work, we provide a much simpler proof of that result and we extend it to access functions with real values.

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Section: Conclusion and Open Problemsmentioning

confidence: 99%

On the Information Ratio of Non-perfect Secret Sharing Schemes

et al. 2016

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“…On the one hand, our main result shows that Csirmaz's method is less powerful than ours for threshold access structures. On the other hand, the alphabet size lower bound of 4 for the tree access structure {01, 12, 03, 34, 05} obtained by Csirmaz and Tardos [21] cannot be matched by our game analysis.…”

Section: Discussionmentioning

confidence: 63%

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Bogdanov¹,

Guo²,

Komargodski³

2020

Theory of Comput.

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We prove that for every n and 1 < t < n any tout of -n threshold secret sharing scheme for one-bit secrets requires share size log(t + 1). Our bound is tight when t = n − 1 and n is a prime power. In 1990 Kilian and Nisan proved the incomparable bound log(n − t + 2). Taken together, the two bounds imply that the share size of Shamir's secret sharing scheme (Comm. ACM 1979) is optimal up to an additive constant even for one-bit secrets for the whole range of parameters 1 < t < n. More generally, we show that for all 1 < s < r < n, any ramp secret sharing scheme with secrecy threshold s and reconstruction threshold r requires share size log((r + 1)/(r − s)). As part of our analysis we formulate a simple game-theoretic relaxation of secret sharing for arbitrary access structures. We prove the optimality of our analysis for threshold secret sharing with respect to this method and point out a general limitation.

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“…On one hand, the information ratios of several small graphs are determined, like graphs up to six vertices [3,4,11,13,19,20] and of some graphs with at most ten vertices [10,12,15,16,21]. On the other hand, in addition to the case of small graphs, exact values for the information ratios of some general families of graphs are proved as well, like hypercubes and d-dimensional lattices [5], trees [9], recursive constructions [2], special unicyclic graphs [12], graphs with large girth [8] or graphs with large girth and no adjacent vertices of high-degree [7].…”

Section: Related Work and Our Contributionmentioning

confidence: 99%

On the Information Ratio of Graphs with Many Leaves

Gyarmati

Ligeti

2019

Tatra Mountains Mathematical Publications

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We investigate the information ratio of graph-based secret sharing schemes. This ratio characterizes the efficiency of a scheme measured by the amount of information the participants must remember for each bits in the secret. We examine the information ratio of several systems based on graphs with many leaves, by proving non-trivial lower and upper bounds for the ratio. On one hand, we apply the so-called entropy method for proving that the lower bound for the information ratio of n-sunlet graphs composed of a 1-factor between the vertices of a cycle Cn and n independent vertices is 2. On the other hand, some symmetric and recursive constructions are given that yield the upper bounds. In particular, we show that the information ratio of every graph composed of a 1-factor between a complete graph Kn and at most 4 independent vertices is smaller than 2.

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Optimal Information Rate of Secret Sharing Schemes on Trees

Cited by 21 publications

References 17 publications

On the Information Ratio of Non-perfect Secret Sharing Schemes

On the Information Ratio of Non-perfect Secret Sharing Schemes

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On the Information Ratio of Graphs with Many Leaves

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