Abstract. In this paper, we show that there exists a &cheater identifiable (k, n) threshold secret sharing scheme such as follows for cheating probability e > 0. If k 2 31 + 1, then 1. Just k participants are enough to identify who are cheaters.2. 1x1 is independent of n. That is, 1x1 = lSl(l/e)(t+2), where S denotes the set of secrets and V, denotes the set of shares of a participant Pi, respectively.(Previously, no schemes were known which satisfy both requirements.)Further, we present a lower bound on lvtl for our model and for the model of Tompa and Woll. Our bound for the TW model is much more tight than the previous bound.