# Towards Characterizing when Information-Theoretic Key Agreement Is Possible

## Ueli Maurer and Stefan Wolf

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```This paper is concerned with information-theoretically secure
secret key agreement in the general scenario where three parties,
Alice, Bob, and Eve, know random variables X, Y, and Z,
respectively, with joint distribution P_XYZ, for instance
resulting from receiving a sequence of random bits broadcast by
a satellite. We consider the problem of determining for a given
distribution P_{XYZ} whether Alice and Bob can in principle,
by communicating over an insecure channel accessible to Eve,
generate a secret key about which Eve's information is arbitrarily
small. When X, Y, and Z are random variables that result from a binary
random variable being sent through three arbitrary independent channels,
it is shown that secret key agreement is possible if and only if I(X;Y|Z)>0,
i.e., under the sole condition that X and Y have some (arbitrarily weak)
statistical dependence when given Z.