Combinatorial Repairability for Threshold Schemes

In this paper, we consider methods whereby a subset of players in a $(k,n)$-threshold scheme can ``repair\u27\u27 another player\u27s 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, and it should be efficient, where efficiency is measured in terms of the amount of information exchanged during the repairing process. We study two approaches to repairing. The first method is based on the ``enrollment protocol\u27\u27 from (Nojoumian-Stinson-Grainger, 2010) which was originally developed to add a new player to a threshold scheme (without the participation of the dealer) after the scheme was set up.The second method distributes ``multiple shares\u27\u27 to each player, as defined by a suitable combinatorial design. This method results in larger shares, but lower communication complexity, as compared to the first method

Designing Efficient Algorithms for Combinatorial Repairable Threshold Schemes

Repairable secret sharing schemes are secret sharing schemes where, without the original dealer who distributed the shares, the participants can combine information from their shares to perform a computation that reconstructs a share for a participant who has lost their share. In this work, we study the repairability of a threshold scheme with respect to the probability that it is possible to perform a repair for a failed share, where each participant in the scheme is available with some probability p. We measure the repairability of a scheme in terms of probability that a repair set is available and in terms of the expected number of available repair sets. Additionally, we design efficient algorithms for determining who to contact when attempting to perform a repair on a failed share for repairable threshold schemes which use 2-designs. We also introduce the use of t-designs, for t > 2, as distribution designs to produce repairable secret sharing schemes with higher repairing degrees and we discuss modifications to the algorithm to account for the different attributes of the designs where t > 2