A network reliability approach to the analysis of combinatorial repairable threshold schemes

Abstract

A repairable threshold scheme (which we abbreviate to RTS ) is a (τ, n)-threshold scheme in which a subset of players can "repair" another player's share in the event that their share has been lost or corrupted. This will take place without the participation of the dealer who set up the scheme. The repairing protocol should not compromise the (unconditional) security of the threshold scheme. Combinatorial repairable threshold schemes (or combinatorial RTS ) were recently introduced by Stinson and Wei [8]. In these schemes, "multiple shares" are distributed to each player, as defined by a suitable combinatorial design called the distribution design. In this paper, we study the reliability of these combinatorial repairable threshold schemes in a setting where players may not be available to take part in a repair of a given player's share. Using techniques from network reliability theory, we consider the probability of existence of an available repair set, as well as the expected number of available repair sets, for various types of distribution designs. * D.R. Stinson's Research is supported by NSERC discovery grant RGPIN-03882.Corollary 3.4. A 3-(v, k, 1)-design can be used as a distribution design to produce an RTS with threshold τ if k ≥ τ (τ − 1) + 1.Remark 3.5. In order to obtain τ = 3, we require k ≥ 7 in Corollary 3.4; to obtain τ = 4, we require k ≥ 13, etc. ReliabilityIn our analysis, to compute the reliability metrics for repair sets, we employ the use of cutsets from network reliability theory (see Colbourn [2] for basic results and terminology relating to network reliability). When using BIBDs as distribution designs, we were able to easily compute reliability formulas in Section 2 without the use of this methodology because the sets C j were disjoint. However, it is advantageous to use cutsets to analyze the reliability of the RTS constructed using distribution designs with t ≥ 3.In this section, for brevity, we will conflate the notion of players and blocks and express all our arguments in terms of blocks of the distribution design (X, B).Definition 3.6. A cutset for a block B is a minimal subset of blocks B ′ such that a repair is not possible if all the blocks in B ′ are not available. A cutset fails if every block in the cutset is not available.Lemma 3.7. Let B = {x 1 , . . . , x k } be a block in the distribution design. Then the sets C j , for 1 ≤ j ≤ k, are the cutsets. Example 3.8. Here are the blocks in an 3-(8, 4, 1)-design: A 1 = {1, 2, 3, 4} A 2 = {5, 6, 7, 8} B 1 = {1, 2, 5, 6} B 2 = {1, 2, 7, 8} B 3 = {1, 3, 5, 7} B 4 = {1, 3, 6, 8} B 5 = {1, 4, 5, 8} B 6 = {1, 4, 6, 7} B 7 = {3, 4, 7, 8} B 8 = {3, 4, 5, 6} B 9 = {2, 4, 6, 8} B 10 = {2, 4, 5, 7} B 11 = {2, 3, 6, 7} B 12 = {2, 3, 5, 8} Suppose A 1 wants to repair their share. Then, the relevant cutsets are

New protocols of cognitive data management and sharing in cloud computing

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“…The processes of data securitization and their safe management are the subject of various analyses, for example, of safe management of selectively chosen data or entire information sets, specifying the need to conceal data, protocols guaranteeing data security, as well as algorithms implementing data securitization processes. Publications describe various examples of data securitization protocols . The algorithms of data splitting and sharing are a special example of protocols used for the processes of data securitization .…”