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 $\ell > 2$, the longest shared secret that can be established from a tuple of strings $(x_1, . . . , x_\ell)$ by $\ell$
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.

Fixed a few minor errors not affecting the main result. Improved presentation

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 $\ell > 2$, the longest shared secret that can be established from a tuple of strings $(x_1, . . . , x_\ell)$ by $\ell$
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.

The presentation is improved. There is no technical change.

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 $\ell > 2$, the longest shared secret that can be established from a tuple of strings $(x_1, . . . , x_\ell)$ by $\ell$
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.