We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties—one having x and the complexity profile of the pair and the other one having y and the complexity profile of the pair—can establish via a probabilistic protocol with interaction on a public channel. For ℓ > 2, the longest shared secret that can be established from a tuple of strings ( x 1 , …, x ℓ ) by ℓ parties—each one having one component of the tuple and the complexity profile of the tuple—is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length for protocols with public randomness. We also show that if the communication complexity drops below the established threshold, then only very short secret keys can be obtained.

中文翻译：

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算法信息论中互信息的操作表征

我们证明了任何一对字符串的互信息，在 Kolmogorov 复杂性的意义上X和是的在对数精度上等于两方最长共享密钥的长度——一方拥有X以及该对的复杂性概况，另一个具有是的以及该对的复杂性概况——可以通过概率协议在公共渠道上进行交互建立。对于 ℓ > 2，可以从字符串元组 (X 1, …,X ℓ) 由 ℓ 方组成——每一方都有元组的一个组成部分和元组的复杂度分布——在对数精度上等于元组的复杂度减去将元组分发给所有各方所需的最小通信量。我们建立了密钥协商协议的通信复杂性，该协议为具有公共随机性的协议生成最大长度的密钥。我们还表明，如果通信复杂度低于既定阈值，则只能获得非常短的密钥。