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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 \( t \)-coverings, which play key roles in these algorithms, is also provided.