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：我们考虑一种秘密分享模式,即经销商在一组参与者之间分配秘密股份,但限制只有预定的参与者子集才能通过共享股份来重建秘密。我们的研究将Shamir的秘密分享模式分为三个方向。首先,我们允许联合设计创建股票和分配股份的协议,而不是将模式限制在独立设计上。第二,我们不假定参与者和经销商能够免费获得信息-理论安全渠道,而是假设他们能够进入公共渠道和相关的随机性。第三,受无线网络设置的驱动,其中相关随机性来自频道获取量的测量,我们探索一个交易商是多个次交易商组成的实体的环境。我们的主要成果是内外部区域,交易商和参与者可以在这一模式中获得可实现的秘密利率。为此,我们开发了两种新的可实现性技术,一种是免费处理分配环境中的可靠性和安全限制,另一种是将每个交易商的无线性网络设置为多交易商的随机性网络设置,在每一个单一交易商的获取能力之间,一种是将每个交易商和每个交易商的互通性标准设定一个多端标准,我们每个交易商的连带的连锁的连锁的连锁的连锁交易商结构,在每一个交易商的进入每个交易商的进入每个交易商的门槛的门槛上。