It is known that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal to the length of the longest shared secret key that two parties can establish via a probabilistic protocol with interaction on a public channel, assuming that the parties hold as their inputs x and y respectively. We determine the worst-case communication complexity of this problem for the setting where the parties can use private sources of random bits. We show that for some x, y the communication complexity of the secret key agreement does not decrease even if the parties have to agree on a secret key whose size is much smaller than the mutual information between x and y. On the other hand, we discuss examples of x, y such that the communication complexity of the protocol declines gradually with the size of the derived secret key. The proof of the main result uses spectral properties of appropriate graphs and the expander mixing lemma, as well as information theoretic techniques.

翻译：已知在柯尔莫哥洛夫复杂性意义下，任意字符串对x和y的互信息等于双方通过公共信道上的交互式概率协议所能建立的最长共享秘密密钥的长度，前提是双方分别持有x和y作为输入。我们确定了在双方可使用私有随机比特源的情况下该问题的最坏情况通信复杂度。研究表明，对于某些x和y，即使双方需协商的秘密密钥尺寸远小于x与y之间的互信息，其秘密密钥协商的通信复杂度也不会降低。另一方面，我们讨论了某些x和y的示例，其中协议的通信复杂度会随生成秘密密钥尺寸的增大而逐步递减。主要结果的证明利用了适当图的谱性质、展开图混合引理以及信息论技术。