Efficient (k,n)Threshold secret sharing schemes secure against cheating from n-1 cheaters

In (k, n) threshold secret sharing scheme, Tompa and Well consider a problem of cheaters who try to make another participant reconstruct invalid secret. Later, the model of such cheating is formalized in some researches. Some schemes secure against cheating of these models are proposed. However, in these models, the number of colluding participants is restricted to k - 1 or less. In this paper, we consider k or more colluding participants. Of course, secrecy is not maintained to such participants. However, if considering detecting the fact of cheating, we need to consider a cheating from k or more colluding participants. In this paper, we propose a (k, n) threshold secret sharing scheme that is capable of detecting the fact of cheating from n - 1 or less colluding participants. A scheme proposed by Tompa and Woll can be proven to be a (k, n) threshold secret sharing scheme that is capable of detecting the fact of cheating from n - I or less colluding participants. However, our proposed scheme is much more efficient with respect to the size of shares.