Towards Attack and Defense Views to (t, n)-threshold Secret Sharing Scheme Using Information Theory

At present, the research on shares in the t-out-of-n secret sharing scheme mainly focuses on the delimitation of share length, and there is no specific quantitative method for the security of authorized shares t in secret sharing without considering any application scenarios. In this paper, we presents a method to quantity the capabilities of the attacker and the defender under the t-out-of-n secret sharing scheme. More specifically, we present a new general notion of limitation to provide a precise bound of attack and defense capability under the $(t,\ n)$-threshold sharing scheme. Furthermore, we discuss the best selection of authorized shares t. Firstly, we transform the attack and defense under the $(t,\ n)$-threshold secret sharing scheme into the communication problem in information theory, and establish the attack channel and defense channel from the point of view of the attacker and the defender. Moreover, we describe the capability of both attacking and defending by introducing average mutual information, and analyze the limitation of the capability of both sides, as well as the maximum value of the average mutual information is the channel capacity. Finally, according to the average amount of mutual information of the change curve under different scenarios of secret sharing, we compare and analyze the influence of the capability limitation of the attacker and the defender on the value of t, and get the best selection of t.