Coding Theorems on the Threshold Scheme for a General Source

In this paper, coding theorems on the (t, m) -threshold scheme for a general source are discussed, where m means the number of the shares and t means a threshold. The (t,m) -threshold scheme treated in this paper encrypts n source outputs Xⁿ to m shares at once and is required to satisfy the two conditions that 1) Xⁿ is reproduced from arbitrary t shares, and 2) almost no information of Xⁿ is revealed from any t - 1 shares. It is shown that the (t,m) -threshold scheme must satisfy certain inequalities including the limit inferiors in probability. One of the inequalities is closely related to the minimum length of the fair random bits needed to a dealer for realizing the (t, m) -threshold scheme. In addition, it is shown that a certain variation of Shamir´s threshold scheme meets the two conditions. The same approach can be taken to the problems of Shannon´s cipher system with the perfect secrecy and fixed-length source coding with vanishing decoding error probability. It is shown that the same kind of inequalities, which indicate the converse coding theorems, hold in both two cases.