Secret sharing with additive access structures from correlated random variables

We generalize secret-sharing models that rely on correlated randomness and public communication, originally designed for a fixed access structure, to support a sequence of dynamic access structures, which we term an Additive Access Structure. Specifically, the access structure is allowed to monotonically grow by having any subset of participants added to it at a given time step, and the dealer only learns of these changes to the access structure on the time step that they occur. For this model, we prove the existence of a secret sharing strategy that achieves the same secret rate at each time step as the best known strategy for the fixed access structure version of this model. We also prove that there exists a strategy that is capacity-achieving at any time step where the access structure is a threshold access structure.