Abstract
Our work deals with secret sharing in the theoretical points of view of Shannon's Information Theory and Kolmogorov's Algorithmic Information Theory. We are going to explain how these three subjects are naturally deeply intertwined. Information inequalities play a central role in this text. They are the inequalities for Shannon entropy, but also they are in exact correspondence with the inequalities for Kolmogorov complexity. Algorithmic Information theory, as introduced by Kolmogorov, formalizes the idea of randomness for strings. These two reasons alone justify to consider the notion of secret sharing in the Algorithmic framework (if one can share a random secret one can share anything). Originally, secret sharing was first studied under the combinatorial lens, only later was it more generally formalized using information-theoretic measures. This step allowed the use of information inequalities which revealed to be very important to understand the existence of secret-sharing schemes with respect to efficiency. The investigation of information inequalities is at its debut. We contribute to the subject by introducing the notion of essentially conditional inequalities, which shows once again that information inequalities are yet not fully understood.