  We address the problem of quantifying the cryptographic content of
probability distributions, in relation to an application to secure multi-party
sampling against a passive t-adversary. We generalize a recently introduced
notion of assisted common information of a pair of correlated sources to that
of K sources and define a family of monotone rate regions indexed by K. This
allows for a simple characterization of all t-private distributions that can be
statistically securely sampled without any auxiliary setup of pre-shared noisy
correlations. We also give a new monotone called the residual total correlation
that admits a simple operational interpretation. Interestingly, for sampling
with non-trivial setups (K > 2) in the public discussion model, our definition
of a monotone region differs from the one by Prabhakaran and Prabhakaran (ITW
2012).
