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Unconditionally Secure Key Agreement and the Intrinsic Conditional Information

Ueli Maurer and Stefan Wolf

This paper is concerned with secret-key agreement by public discussion. Assume that two parties Alice and Bob and an adversary Eve have access to independent realizations of random variables $X$, $Y$, and $Z$, respectively, with joint distribution $P_{XYZ}$. The secret key rate $S(X;Y||Z)$ has been defined as the maximal rate at which Alice and Bob can generate a secret key by communication over an insecure, but authenticated channel such that Eve's information about this key is arbitrarily small. We define a new conditional mutual information measure, the intrinsic\/} conditional mutual information between $X$ and $Y$ when given $Z$, denoted by $\ida$, which is an upper bound on $S(X;Y||Z)$. The special scenarios are analyzed where $X$, $Y$, and $Z$ are generated by sending a binary random variable $R$, for example a signal broadcast by a satellite, over independent channels, or two scenarios in which $Z$ is generated by sending $X$ and $Y$ over erasure channels.In the first two scenarios it can be shown that the secret key rate is strictly positive if and only if $\ida$ is strictly positive. For the third scenario a new protocol is presented which allows secret-key agreement even when all the previously known protocols fail.