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Coding Theorems on the Threshold Scheme for a General Source

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1 Author(s)
Koga, H. ; Grad. Sch. of Syst. & Inf. Eng., Univ. of Tsukuba, Tsukuba

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.

Published in:

Information Theory, IEEE Transactions on  (Volume:54 ,  Issue: 6 )