An application of covering designs: determining the maximum consistent set of shares in a threshold scheme

The shares in a (k, n) Shamir threshold scheme consist of n points on some polynomial of degree at most k − 1. If one or more of the shares are faulty, then the secret may not be reconstructed correctly. Supposing that at most t of the n shares are faulty, we show how a suitably chosen covering design can be used to compute the correct secret. We review known results on coverings of the desired type, and give some new constructions. We also consider a randomized algorithm for the same problem, and compare it with the deterministic algorithm obtained by using a particular class of coverings. ∗research supported by NSERC grant OGP # 333195 †research supported by NSERC grants IRC # 216431-96 and RGPIN # 203114-98 ‡research supported by NSERC grant OGP # 0003558