We consider a secret-sharing model where a dealer distributes the shares of a secret among a set of participants with the constraint that only predetermined subsets of participants must be able to reconstruct the secret by pooling their shares. Our study generalizes Shamir's secret-sharing model in three directions. First, we allow a joint design of the protocols for the creation of the shares and the distribution of the shares, instead of constraining the model to independent designs. Second, instead of assuming that the participants and the dealer have access to information-theoretically secure channels at no cost, we assume that they have access to a public channel and correlated randomness. Third, motivated by a wireless network setting where the correlated randomness is obtained from channel gain measurements, we explore a setting where the dealer is an entity made of multiple sub-dealers. Our main results are inner and outer regions for the achievable secret rates that the dealer and the participants can obtain in this model. To this end, we develop two new achievability techniques, a first one to successively handle reliability and security constraints in a distributed setting, and a second one to reduce a multi-dealer setting to multiple single-user dealer settings. Our results yield the capacity region for threshold access structures when the correlated randomness corresponds to pairwise secret keys shared between each sub-dealer and each participant, and the capacity for the all-or-nothing access structure in the presence of a single dealer and arbitrarily correlated randomness.

翻译：我们考虑一种秘密共享模型，其中分发者将秘密的份额分配给一组参与者，但约束条件是只有预先指定的参与者子集才能通过汇集其份额重构秘密。我们的研究在三个方向上扩展了Shamir的秘密共享模型。首先，我们允许联合设计份额生成与份额分发的协议，而非将模型限制为独立设计。其次，我们不再假设参与者和分发者可零成本使用信息论安全信道，而是假设他们可以访问公开信道和相关随机性。第三，受无线网络场景（其中相关随机性源自信道增益测量）的启发，我们探索了分发者由多个子分发者组成的实体情形。我们的主要成果是给出了该模型下分发者与参与者可获得的可达秘密速率的内外区域。为此，我们开发了两种新的可达性技术：第一种技术用于在分布式设置中依次处理可靠性与安全约束，第二种技术将多分发者设置简化为多个单用户分发者设置。当相关随机性对应于每个子分发者与每个参与者之间共享的成对密钥时，我们的结果为阈值访问结构给出了容量区域；在单个分发者及任意相关随机性存在的情况下，则为全有或全无访问结构给出了容量。