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Cryptography and Communications

Error decodable secret sharing and one-round perfectly secure message transmission for general adversary structures

Keith M. Martin, Maura B. Paterson, Douglas R. Stinson

Published Dec 15, 2010• 16 citations
Publisher
DOI
10.1007/s12095-010-0039-6

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Error Decodable Secret Sharing and One-Round Perfectly Secure Message Transmission for General Adversary Structures.

Keith M. Martin, Maura B. Paterson, Douglas R. Stinson

An error decodable secret-sharing scheme is a secret-sharing scheme with the additional property that the secret can be recovered from the set of all shares, even after a coalition of participants corrupts the shares they possess. In this paper, schemes that can tolerate corruption by sets of participants belonging to a monotone coalition structure are considered. This coalition structure may be unrelated to the authorised sets of the secret-sharing scheme. This is generalisation of both a related notion studied in the context of multiparty computation, and the well-known error-correction properties of threshold schemes based on Reed-Solomon codes. Necessary and sufficient conditions for the existence of such schemes are deduced, and methods for reducing the storage requirements of a technique of Kurosawa for constructing error-decodable secret-sharing schemes with efficient decoding algorithms are demonstrated. In addition, the connection between one-round perfectly secure message transmission (PSMT) schemes with general adversary structures and secret-sharing schemes is explored. We prove a theorem that explicitly shows the relation between these structures. In particular, an error decodable secret-sharing scheme yields a one-round PSMT, but the converse does not hold. Furthermore, we are able to show that some well-known results concerning one-round PSMT follow from known results on secret-sharing schemes. These connections are exploited to investigate factors affecting the performance of one-round PSMT schemes such as the number of channels required, the communication overhead, and the efficiency of message recovery.

•Jan 1, 2009•0 citations

General Error Decodable Secret Sharing Scheme and Its Application

Kaoru Kurosawa

Consider a model of secret sharing schemes with cheaters. We say that a secret sharing scheme is error decodable if we can still recover the secretscorrectly from a noisy share vector (share1', ..., sharen'). In this paper, we first prove that a perfect secret sharing scheme is error decodable if and only if the adversary structure Γ satisfies a certain condition calledQ3. Next, for such Γ , we show a scheme such that the decoding algorithm runs in polynomial-time in |S| and the size of a linear secret sharing scheme which realizes Γ. We finally show an application to 1-round perfectly secure message transmission schemes (PSMT).

IEEE Transactions on Information Theory•Sep 1, 2011•8 citations
An error decodable secret-sharing scheme is a secret-sharing scheme with the additional property that the secret can be recovered from the set of all shares, even after a coalition of participants corrupts the shares they possess. In this paper, schemes that can tolerate corruption by sets of participants belonging to a monotone coalition structure are considered. This coalition structure may be unrelated to the authorised sets of the secret-sharing scheme. This is generalisation of both a related notion studied in the context of multiparty computation, and the well-known error-correction properties of threshold schemes based on Reed-Solomon codes. Necessary and sufficient conditions for the existence of such schemes are deduced, and methods for reducing the storage requirements of a technique of Kurosawa for constructing error-decodable secret-sharing schemes with efficient decoding algorithms are demonstrated. In addition, the connection between one-round perfectly secure message transmission (PSMT) schemes with general adversary structures and secret-sharing schemes is explored. We prove a theorem that explicitly shows the relation between these structures. In particular, an error decodable secret-sharing scheme yields a one-round PSMT, but the converse does not hold. Furthermore, we are able to show that some well-known results concerning one-round PSMT follow from known results on secret-sharing schemes. These connections are exploited to investigate factors affecting the performance of one-round PSMT schemes such as the number of channels required, the communication overhead, and the efficiency of message recovery.
Consider a model of secret sharing schemes with cheaters. We say that a secret sharing scheme is error decodable if we can still recover the secretscorrectly from a noisy share vector (share1', ..., sharen'). In this paper, we first prove that a perfect secret sharing scheme is error decodable if and only if the adversary structure Γ satisfies a certain condition calledQ3. Next, for such Γ , we show a scheme such that the decoding algorithm runs in polynomial-time in |S| and the size of a linear secret sharing scheme which realizes Γ. We finally show an application to 1-round perfectly secure message transmission schemes (PSMT).