Paper 2019/598

Partial Secret Sharing

Amir Jafari, Reza Kaboli, and Shahram Khazaei

Abstract

Information ratio of an access structure is an important measure for efficiency of the best secret sharing scheme realizing it. The most common notion of secret sharing security is that of total (perfect) realization. Two well-known relaxations are the notions of statistical and quasi-total secret sharing. In this paper, we study the relation between different security notions. The most significant and technical result of this paper is that quasi-total and total information ratios coincide for linear schemes. To this end, we employ some tools from linear algebra in companion with a newly introduced relaxed security notion, called partial realization. We provide some intuition that why proving coincidence/separation between total and quasi-total information ratios for the class of abelian schemes is probably much more challenging. We also present some additional results which shed further light on our understanding of different security notions. In particular, one of our results, in combination with a recent result, shows that statistical and total security notions coincide for the class of group-homomorphic schemes, or maybe even a larger class.

Note: Some minors corrections. In particular Remark 1.1 and Remark 1.2 have been added.