On Algorithms for Searching a Consistent Set of Shares in a Threshold Scheme and the Related Covering Problem

摘要

In the Shamir (k, n) threshold scheme, if one or more of the n shares are fake, then the secret may not be reconstructed correctly by some sets of k shares. Supposing that at most t of the n shares are fake, Rees et al. (1999) described two algorithms to determine consistent sets of shares so that the secret can be reconstructed correctly from k shares in any of these consistent sets. In their algorithms, no honest participant can be absent and at least n - t shares should be pooled during the secret reconstruction phase. In this paper, we propose a modified algorithm for this problem so that the number of participants taking part in the secret reconstruction can be reduced to k + 2t and the shares need to be pooled can be reduced to, in the best case, k + t, and less than or equal to k + 2t in the others. Its performance is evaluated. A construction for t-covermgs, which play key roles in these algorithms, is also provided.