Secret sharing with additive access structures from correlated random variables
David Miller, Rémi A. Chou
TL;DR
This paper extends secret sharing with correlated randomness and public communication to Additive Access Structures (AAS), where authorized sets can grow over time and the dealer learns changes only at each step. It proves an achievability bound that yields a per-time-step secret rate $R_t = \min_{\mathcal{U}\in\mathbb{U}_t} H(Y|X_{\mathcal{U}}) - \max_{\mathcal{A}\in\mathbb{A}_t} H(Y|X_{\mathcal{A}})$, and a matching converse, showing the rate depends only on the current access structure and source distribution. For TAAS, the results show capacity $R_t = H(X)(u-v)$ under a specific modeling of the source and keys, indicating no advantage in knowing the AAS in advance. The achievability uses a novel quantized random-binning scheme that is valid a priori for all possible evolutions of the AAS, ensuring reliability and secrecy across time steps. These findings generalize fixed-structure secret sharing and reveal fundamental limits for dynamic access-control in correlated randomness settings with public communication.
Abstract
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.