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

Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory