In multi-party key agreement protocols it is assumed that the parties are given correlated input data and should agree on a common secret key so that the eavesdropper cannot obtain any information on this key by listening to the communications between the parties. We consider the one-shot setting, when there is no ergodicity assumption on the input data. It is known that the optimal size of the secret key can be characterized in terms of the mutual information between different combinations of the input data sets, and the optimal key can be produced with the help of the omniscience protocol. However, the optimal communication complexity of this problem remains unknown. We show that the communication complexity of the omniscience protocol is optimal, at least for some complexity profiles of the input data, in the setting with restricted interaction between parties (the simultaneous messages model). We also provide some upper and lower bounds for communication complexity for other communication problems. Our proof technique combines information-theoretic inequalities and the spectral method.

翻译：在多方密钥协商协议中，假设各方接收到相关的输入数据，应协商出一个共同的密钥，使得窃听者无法通过监听各方之间的通信获得该密钥的任何信息。我们考虑单次设置，即不对输入数据做遍历性假设。已知密钥的最优大小可以通过不同输入数据集组合之间的互信息来刻画，且可通过全知协议生成最优密钥。然而，该问题的最优通信复杂度仍未知。我们证明，在各方交互受限的设置（同时消息模型）下，至少对于输入数据的某些复杂度分布，全知协议的通信复杂度是最优的。我们还为其他通信问题的通信复杂度提供了一些上界和下界。我们的证明技术结合了信息论不等式和谱方法。