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.