Multiple Threshold Schemes Under The Weak Secure Condition

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.