In this work we revisit the fundamental findings by Chen et al. in [5] on general information transfer in linear ramp secret sharing schemes to conclude that their method not only gives a way to establish worst case leakage [5, 25] and best case recovery [5, 19], but can also lead to additional insight on non-qualifying sets for any prescribed amount of information. We then apply this insight to schemes defined from monomial-Cartesian codes and by doing so we demonstrate that the good schemes from Sec.\ IV in [14] have a second layer of security. Elaborating further, when given a designed recovery number, in a new construction the focus is entirely on ensuring that the access structure possess desirable second layer security, rather on what is the worst case information leakage in terms of number of participants. The particular structure of largest possible sets being not able to determine given amount of information suggests that we call such schemes democratic

翻译：本文重新审视了Chen等人在文献[5]中关于线性斜坡秘密共享方案中一般信息传输的基础性发现，并指出他们的方法不仅能建立最坏情况泄漏[5,25]和最佳情况恢复[5,19]，还能为任意预设信息量下的非授权集合提供额外洞见。随后，我们将这一洞见应用于由单项式笛卡尔码定义的方案，并藉此证明文献[14]第IV节中的优良方案具有第二层安全性。进一步阐释：当给定预设恢复阈值时，新构建方案完全聚焦于确保访问结构具备理想的第二层安全性，而非关注以参与者数量衡量的最坏情况信息泄漏。最大可能集合无法确定特定信息量的特殊结构特征，促使我们将此类方案命名为民主式方案。