多终端信源模型中实现SK容量所需的公共通信
On the Public Communication Needed to Achieve SK Capacity in the Multiterminal Source Model
摘要：
本文的重点是在Csisz{á}r和Narayan的多终端信源模型中产生最大速率秘密密钥（SK）所需的公共通信。基于Tyagi先前的工作，为两个终端的情况下，我们推导出一个下界的通信复杂性，$R_{\text{SK}}$，定义为最低速率的公共通信需要产生一个最大速率SK。众所周知，最低速率的通信全知，由$R_{\text{CO}}$表示，是上界$R_{\text{SK}}$。对于一类定义在一致超图上的两两独立网络（PIN）模型，我们证明了一个在多项式时间内可验证的“$\mathcal{S}$型“条件，保证了我们关于$R_{\text{SK}}$的下界满足$R_{\text{CO}}$的上界.因此，满足我们条件的PIN模型是$R_{\text{SK}}$ -极大的，这意味着上界$R_{\text{SK}} \le R_{\text{CO}}$等式成立。这使我们能够明确评估$R_{\text{SK}}$的PIN模型。我们还给出了几个满足$\mathcal S$型条件的PIN模型的例子。最后，我们证明了，对于任意的多终端源模型，我们的类型$\mathcal S$条件的严格版本意味着通信从\emph{all}终端（“多功能”）是需要建立一个SK的最大速率。对于三端源模型，反之亦然：只有当满足严格的$\mathcal S$型条件时，才需要多声性来生成最大速率SK。存在的反例表明，相反的是不正确的，在一般情况下，源模型与四个或更多的终端。
英文摘要：
The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csisz{á}r and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, $R_{\text{SK}}$, defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by $R_{\text{CO}}$, is an upper bound on $R_{\text{SK}}$. For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain "Type $\mathcal{S}$" condition, which is verifiable in polynomial time, guarantees that our lower bound on $R_{\text{SK}}$ meets the $R_{\text{CO}}$ upper bound. Thus, PIN models satisfying our condition are $R_{\text{SK}}$-maximal, meaning that the upper bound $R_{\text{SK}} \le R_{\text{CO}}$ holds with equality. This allows us to explicitly evaluate $R_{\text{SK}}$ for such PIN models. We also give several examples of PIN models that satisfy our Type $\mathcal S$ condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type $\mathcal S$ condition implies that communication from \emph{all} terminals ("omnivocality") is needed for establishing a SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type $\mathcal S$ condition is satisfied. Counterexamples exist that show that the converse is not true in general for source models with four or more terminals.
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