Partial Secret Sharing Schemes

The information ratio of an access structure is an important parameter for quantifying the efficiency of the best secret sharing scheme (SSS) realizing it. The most common security notion is perfect security. The following relaxations, in increasing level of security, have been presented in the literature: quasi-perfect, almost-perfect and statistical. Understanding the power of relaxing the correctness and privacy requirements in the efficiency of SSSs 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. We refer to this gap as the nominal capacity. We quantify the efficiency of such schemes using a parameter called partial information ratio. It is defined to be the same as the (standard) information ratio, except that we divide the largest share entropy by nominal capacity instead of the secret entropy. Despite this modification, partial security turns out weaker than the weakest mentioned non-perfect security notion, i.e., quasi-perfect security. We present three main results in this paper. First, we prove that partial and perfect information ratios coincide for the class of linear SSSs. Consequently, for this class, information ratio is invariant with respect to all security notions. Second, by viewing a partial SSS as a wiretap channel, we prove that for the general (i.e., non-linear) class of SSSs, partial and statistical information ratios are equal. Consequently, for this class, information ratio is invariant with respect to all non-perfect security notions. 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). Our first result strengthens the previous decomposition theorems for constructing perfect linear schemes. Our second result leads to a very strong decomposition theorem for constructing general (i.e., non-linear) statistical schemes. Our third result provides a rare example of the effect of imperfection on the efficiency of SSSs for a certain class of schemes

Interdisciplinary elucidation of concepts, metaphors, theories and problems concerning INFORMATION

Terms included in this glossary recap some of the main concepts, theories, problems and metaphors concerning INFORMATION in all spheres of knowledge. This is the first edition of an ambitious enterprise covering at its completion all relevant notions relating to INFORMATION in any scientific context. As such, this glossariumBITri is part of the broader project BITrum, which is committed to the mutual understanding of all disciplines devoted to information across fields of knowledge and practice. This glossary pretends to make explicit the conflicts and agreements among use and meaning of terms related to information phenomena. Information is approached from opposing paradigms and also from competing and cooperating disciplines. Both in science and in ordinary life, conceptual, ethical, technical and societal problems regard information in an essential way. This glossary does not endorse or presuppose any paradigm or any theory, but rather locates into a public, explicit and commonly understandable space some of the crucial assumptions dividing informational concepts, theories, problems and metaphors. Moreover, we purport to embrace all distinct paradigms with a critical and comprehensive attitude. The glossary is the result of an original methodology, which places any entrance under the responsibility of its editor. Authors possibly distinct from the editor contribute to different articles with texts, comments or discussions. Since authors come from many distinct fields of knowledge, each article should reflect many perspectival but rigorous approaches. The glossary is an open work: the number and contents of all its entrances are updated and submitted to revision by editors and authors. For this reason, this first edition is only a first step in the active development of this collaborative methodology. Any interested reader wishing to contribute, may contact the general editors. This glossary is most indebted to the enthusiasm and work of José María Díaz Nafría. The editorial team, authors and correctors thank the Universidad de León and Caja España for their support to this initiative