We propose a method to study the secrecy constraints in key generation problems where side information might be present at untrusted users. Our method is inspired by a recent work of Hayashi and Tan who used the Rényi divergence as the secrecy measure to study the output statistics of applying hash functions to a random sequence. By generalizing the achievability result of Hayashi and Tan to the multi-terminal case, we obtain the output statistics of applying hash functions to multiple random sequences, which turn out to be an important tool in the achievability proof of strong secrecy capacity regions of key generation problems with side information at untrusted users. To illustrate the power of our method, we derive the capacity region of the multiple private key generation problem with an untrusted helper for continuous memoryless sources under Markov conditions. The converse proof of our result follows by generalizing a result of Nitinawarat and Narayan to the case with side information at untrusted users.

中文翻译：

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使用助手为连续无记忆源生成多个私钥

我们提出了一种方法来研究密钥生成问题中的保密约束，在这些问题中，边信息可能出现在不受信任的用户处。我们的方法受到Hayashi和Tan最近的工作的启发，他们使用Rényi散度作为保密措施来研究将哈希函数应用于随机序列的输出统计量。通过将Hayashi和Tan的可实现性结果推广到多终端情况，我们获得了将哈希函数应用于多个随机序列的输出统计信息，这证明它是密钥生成强保密能力区域可实现性证明的重要工具在不受信任的用户处获取辅助信息的问题。为了说明我们方法的力量，我们在马尔可夫条件下，使用连续的无记忆源的不受信任的辅助方法，得出了多个私钥生成问题的容量区域。我们的结果的相反证明是将Nitinawarat和Narayan的结果推广到带有不信任用户的附带信息的情况下。