Multiple Threshold Schemes Under The Weak Secure Condition
CoRR(2023)
摘要
In this paper, we consider the case that sharing many secrets among a set of
participants using the threshold schemes. All secrets are assumed to be
statistically independent and the weak secure condition is focused on. Under
such circumstances we investigate the infimum of the (average) information
ratio and the (average) randomness ratio for any structure pair which consists
of the number of the participants and the threshold values of all secrets.
For two structure pairs such that the two numbers of the participants are the
same and the two arrays of threshold values have the subset relationship, two
leading corollaries are proved following two directions. More specifically, the
bound related to the lengths of shares, secrets and randomness for the complex
structure pair can be truncated for the simple one; and the linear schemes for
the simple structure pair can be combined independently to be a multiple
threshold scheme for the complex one. The former corollary is useful for the
converse part and the latter one is helpful for the achievability part.
Three new bounds special for the case that the number of secrets
corresponding to the same threshold value $ t $ is lager than $ t $ and two
novel linear schemes modified from the Vandermonde matrix for two similar cases
are presented. Then come the optimal results for the average information ratio,
the average randomness ratio and the randomness ratio. We introduce a tiny
example to show that there exists another type of bound that may be crucial for
the information ratio, to which we only give optimal results in three cases.
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