Partial Secret Sharing Schemes

The following standard relaxations of perfect security for secret sharing schemes (SSSs) exist in the literature: quasi-perfect, almost-perfect, and statistical. Understanding the power of these relaxations on the efficiency of SSSs, measured via a parameter called information ratio, is a long-standing open problem. In this article, we introduce and study an extremely relaxed security notion, called partial security, for which it is only required that any qualified set gains strictly more information about the secret than any unqualified one. To get a meaningful efficiency measure, we normalize the (standard) information ratio of such schemes by an appropriate parameter and refer to the new measure as partial information ratio. We present three main results in this paper. First, we prove that partial and perfect information ratios coincide for the class of linear SSSs. Second, we prove that for the general (i.e., non-linear) class of SSSs, partial and statistical information ratios are equal. Third, we show that partial and almost-perfect information ratios do not coincide for the class of mixed-linear schemes (i.e., schemes constructed by combining linear schemes with different underlying finite fields). We also use the notion of partial secret sharing to strengthen and unify the previous decomposition theorems for constructing SSSs.