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.

翻译：本文考虑使用阈值方案在一组参与者之间共享多个秘密的情况。假设所有秘密统计独立，并重点关注弱安全条件。在此情形下，我们研究任意由参与者数量与所有秘密的阈值构成的结构对中（平均）信息率与（平均）随机性率的下确界。针对两个参与者数量相同且阈值数组具有子集关系的结构对，我们从两个方向证明了两条关键推论。具体而言：复杂结构对中与份额、秘密及随机性长度相关的界可截断为简单结构对的对应界；同时，针对简单结构对的线性方案可独立组合成复杂结构对的多重阈值方案。前者推论对逆命题部分有效，后者推论有助于构造可实现性部分。针对同一阈值t对应的秘密数量大于t的特殊情形，我们提出了三个新界，并给出了两种基于范德蒙德矩阵改进的线性方案来应对两种相似情形。进而给出了（平均）信息率、（平均）随机性率及随机性率的最优结果。最后通过一个简单实例表明，存在另一类可能对信息率至关重要的界，对此我们仅在三种情形下给出最优结果。