Secret sharing is an instrumental tool for sharing secret keys in distributed systems. In a classical threshold setting, this involves a dealer who has a secret/key, a set of parties/users to which shares of the secret are sent, and a threshold on the number of users whose presence is needed in order to recover the secret. In secret sharing, secure links with no leakage are often assumed between the involved parties. However, when the users are nodes in a communication network and all the links are physical links, e.g., wireless, such assumptions are not valid anymore. In order to study this critical problem, we propose a statistical leakage model of secret sharing, where some noisy versions of all the secret shares might be independently leaked to an adversary. We then study the resilience of the seminal Shamir's secret sharing scheme with statistical leakage, and bound certain measures of security (i.e., semantic security, mutual information security), given other parameters of the system including the amount of leakage from each secret share. We show that for an extreme scenario of Shamir's scheme, in particular when the underlying field characteristic is $2$, the security of each bit of the secret against leakage improves exponentially with the number of users. To the best of our knowledge, this is the first attempt towards understanding secret sharing under general statistical noisy leakage.

翻译：秘密共享是分布式系统中共享密钥的重要工具。在经典的阈值设置中，涉及一个拥有秘密/密钥的分配者、一组接收秘密份额的参与方/用户，以及恢复秘密所需的最小用户数量阈值。在秘密共享中，通常假设参与方之间拥有无泄漏的安全链路。然而，当用户是通信网络中的节点且所有链路均为物理链路（例如无线链路）时，这种假设不再成立。为研究这一关键问题，我们提出了一种秘密共享的统计泄漏模型，其中所有秘密份额的某些带噪声版本可能独立泄漏给敌手。随后，我们研究了具有统计泄漏的经典Shamir秘密共享方案的抵抗性，并根据系统其他参数（包括每个秘密份额的泄漏量）界定了若干安全度量（即语义安全性、互信息安全性）。我们证明，在Shamir方案的极端场景下（特别是当底层域特征为$2$时），秘密每个比特对泄漏的安全性随用户数量呈指数级提升。据我们所知，这是首次尝试理解一般统计噪声泄漏下的秘密共享问题。