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).