# The Strong Secret Key Rate of Discrete Random Triples

## Ueli Maurer

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```Three parties, Alice, Bob and Eve, know the sequences of random
variables $X^N=[X_1,X_2,\ldots X_N]$, $Y^N=[Y_1,Y_2,\ldots Y^N]$ and
$Z^N=[Z_1,Z_2,\ldots Z_N]$, respectively, where the triples
$(X_iY_iZ_i)$, for $1\leq i\leq N$, are generated by a discrete
memoryless source according to some probability distribution $P_{XYZ}$.
Motivated by Wyner's and Csiszá}r and Kö}rner's pioneering
definition of, and work on, the secrecy capacity of a broadcast
channel, the secret key rate of $P_{XYZ}$ was defined by Maurer as the
maximal rate $M/N$ at which Alice and Bob can generate secret shared
random key bits $S_1\pp S_M$ by exchanging messages over an insecure
public channel accessible to Eve, such that the rate at which Eve
obtains information about the key is arbitrarily small, i.e., such that
$\lim_{N\ra\infty}I(S_1\pp S_M;Z^N,C^t)/N=0$, where $C^t$ is the
collection of messages exchanged between Alice and Bob over the public
channel. However, this definition is not completely satisfactory
because only the rate, but not the total amount of information about
the key obtained by Eve is bounded. This paper introduces and
investigates the * strong*} secret key rate: it is required that the
total amount of information about the key obtained by Eve be
negligible, i.e. $\lim_{N\ra\infty}I(S_1\pp S_M;Z^N,C^t)=0$, and that
$[S_1\pp S_M]$ be arbitrarily close to uniformly distributed, i.e.
$\lim_{N\ra\infty}M-H([S_1\pp S_M])=0$. Using novel results on privacy
amplification by Bennett, Brassard, Crépeau and Maurer we demonstrate
that the known results for the secret key rate also hold for the
stronger definition.