On ideal homomorphic secret sharing schemes and their decomposition

In 1992, Frankel and Desmedt introduced a technique that enables one to reduce the secret space of an ideal homomorphic secret sharing scheme (IHSSS) into any of its characteristic subgroups. In this paper, we propose a similar technique to reduce the secret space of IHSSSs called the quotient technique. By using the quotient technique, we show that it is possible to yield an ideal linear scheme from an IHSSS for the same access structure, providing an alternative proof of a recent result by Jafari and Khazaei. Moreover, we introduce the concept of decomposition of secret sharing schemes. We give a decomposition for IHSSSs, and as an application, we present a necessary and sufficient condition for an IHSSS to be mixed-linear. Continuing this line of research, we explore the decomposability of some other scheme classes.