On the Information Ratio of Non-perfect Secret Sharing Schemes

Abstract

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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Correspondence to Oriol Farràs.

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Part of this work was presented in the conference CRYPTO 2014 and appeared in its proceedings [26]. Oriol Farràs is supported by the Spanish Government through a Juan de la Cierva grant, TIN2011C27076-C03-01, TIN2014-57364-C2-1-R, by the European Union through H2020-ICT-2014-1-644024, and by the Government of Catalonia through Grant 2014 SGR 537. Part of the work of Torben B. Hansen was done while at Aarhus University and Universitat Rovira i Virgili. Tarik Kaced is supported in part by a grant from the University Grants Committee of the Hong Kong SAR, China (Project No. AoE/E-02/08), and by EQINOCS ANR 11 BS02 004 03. Carles Padró is supported by the Spanish Government under the project MTM2013-41426-R. Part of this research work was done while Carles Padró was with Nanyang Technological University, Singapore.

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Farràs, O., Hansen, T.B., Kaced, T. et al. On the Information Ratio of Non-perfect Secret Sharing Schemes. Algorithmica 79, 987–1013 (2017). https://doi.org/10.1007/s00453-016-0217-9

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Keywords

  • Secret sharing
  • Non-perfect secret sharing
  • Access function
  • Information ratio
  • Polymatroid