# Unconditionally Secure Key Agreement and the Intrinsic Conditional Information

## Ueli Maurer and Stefan Wolf

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```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.