Coding Theorems on the Threshold Scheme for a General Source

13 years 4 months ago

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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 X n to m shares at once and is required to satisfy the two conditions that 1) X n is reproduced from arbitrary t shares, and 2) almost no information of X n is revealed from any t 01 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 s...

Hiroki Koga

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