Threshold Secret Sharing Scheme for Compartmented Access Structures

Threshold Secret Sharing Scheme for Compartmented Access Structures

P. Mohamed Fathimal, P. Arockia Jansi Rani
Copyright: © 2020 |Pages: 11
DOI: 10.4018/978-1-7998-1763-5.ch026
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In the realm of visual cryptography, secret sharing is the predominant method of transmission and reception of secure data. Most of the (n, n) secret sharing schemes suffer from one common flaw — locking of information when the all- n number of receivers are not available for some reason. This paper proposes a new method of compartmented secret sharing scheme where some threshold number of equally privileged from each compartment can retrieve data. This scheme rules out regeneration of secret image at the single compartment thereby eliminating the danger of misusing secret image. The key features of this scheme are: better visual quality of the recovered image with no pixel expansion; non-requirement of half toning of color images; less computational complexity by reconstructing secret through XORing and simple addition of all share images. This scheme is highly beneficial in applications where data has to be stored securely in a database and in cloud computing to synchronize information passed to different groups or clusters from a single host.
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Many researchers introduced new secret sharing scheme based either on Shamir’s scheme (1979) or with new concepts. Shamir introduced multipartite access structures in his seminal work for weighted threshold access structures. Blakely (1979) introduced geometric threshold secret sharing scheme. Mignotte (1983) and Asmuth-Bloom (1983) developed threshold secret sharing scheme based on the Chinese remainder theorem.

Mignotte’s threshold secret sharing scheme (1983) uses special sequences of integers, referred to as Mignotte sequences. A (k, n)-Mignotte sequence is a sequence of positive integers m1 < · · · < mn such that (mi, mj) = 1, for all 1 ≤ i < j ≤ n, and mn−k+2 · · · mn < m1 · · · mk. The scheme proposed by Asmuth and Bloom (1983) also used special sequences of integers. More exactly, a sequence of pairwise coprime positive integers r, m1 < · · · < mn is chosen such that r · mn−k+2 · · · mn < m1 · · · mk. This scheme was generalized for allowing modules that are not necessarily pairwise coprime in an obvious manner.

Brickell (1990) proposed an elegant solution by choosing the secret S as a combination of m compartment secrets and using a threshold secret sharing scheme for each compartment. In the reconstruction phase, if the number of participants from the jth compartment is greater than or equal to the kj, for all 1 ≤ j ≤ m, then all compartment secrets can be recovered and thus the secret S can be obtained. Brickell proved that all multilevel and compartmented access structures are ideal. He proved that every structure in one of those families admits a vector space secret sharing scheme over every large enough field. Even though the proof is constructive, this scheme did not explain how to construct efficiently.

Simmons (1990) introduced two families of multipartite access structures, the so-called multilevel and compartmented access structures by generalizing the geometrical threshold scheme by Blakely (1979) and he speculated that this was possible for all of them. He has presented the example of an official action that requires at least two Americans and at least two Russians for its initiation.

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