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An Information Theoretic Approach to Secret Sharing

机译：秘密共享的信息理论方法

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摘要

A novel information theoretic approach is proposed to solve the secret sharing problem, in which a dealer distributes one or multiple secrets among a set of participants in such a manner that for each secret only qualified sets of users can recover this secret by pooling their shares together while nonqualified sets of users obtain no information about the secret even if they pool their shares together. While existing secret sharing systems (implicitly) assume that communications between the dealer and participants are noiseless, this paper takes a more practical assumption that the dealer delivers shares to the participants via a noisy broadcast channel. Thus, in contrast to the existing solutions that are mainly based on number theoretic tools, an information theoretic approach is proposed, which exploits the channel randomness during delivery of shares as additional resources to achieve secret sharing requirements. In this way, secret sharing problems can be reformulated as equivalent secure communication problems via wiretap channel models, and can hence be solved by employing the powerful information theoretic security techniques. This approach is first developed for the classic secret sharing problem, in which only one secret is to be shared. This classic problem is shown to be equivalent to a communication problem over a compound wiretap channel. Thus, the lower and upper bounds on the secrecy capacity of the compound channel provide the corresponding bounds on the secret sharing rate, and the secrecy scheme designed for the compound channel provides the secret sharing schemes. The power of the approach is further demonstrated by a more general layered multisecret sharing problem, which is shown to be equivalent to the degraded broadcast multiple-input multiple-output (MIMO) channel with layered decoding and secrecy constraints. The secrecy capacity region for the degraded MIMO broadcast channel is characterized, which provides the secret sharing capacity re- ion. Furthermore, the secure encoding scheme that achieves the secrecy capacity region provides an information theoretic scheme for sharing the secrets.
机译：提出了一种新颖的信息理论方法来解决秘密共享问题，其中交易者以一种方式在一组参与者之间分配一个或多个秘密，使得对于每个秘密，只有合格的用户才能通过将他们的份额集中在一起来恢复该秘密。而不合格的用户集即使将他们的份额集中在一起也不会获得有关机密的信息。尽管现有的秘密共享系统（隐式地）假定交易者与参与者之间的通信是无干扰的，但本文采用了更为实际的假设，即交易者通过嘈杂的广播频道向参与者传递了份额。因此，与主要基于数论工具的现有解决方案相反，提出了一种信息论方法，该方法利用共享交付期间的信道随机性作为额外资源来实现秘密共享需求。这样，秘密共享问题可以通过窃听通道模型重新表示为等效的安全通信问题，因此可以通过采用强大的信息理论安全技术来解决。这种方法是针对经典秘密共享问题而首先开发的，在经典秘密共享问题中，仅一个秘密将被共享。该经典问题显示为等效于复合窃听通道上的通信问题。因此，复合信道的保密容量的上限和下限提供了秘密共享率的相应边界，为复合信道设计的保密方案提供了秘密共享方案。通过更通用的分层多秘密共享问题进一步证明了该方法的功能，该问题被认为等同于具有分层解码和保密约束的降级广播多输入多输出（MIMO）信道。表征了降级的MIMO广播信道的保密容量区域，该区域提供了秘密共享容量区域。此外，实现保密容量区域的安全编码方案提供了用于共享秘密的信息理论方案。

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