Multipartite monotones for secure sampling by public discussion from noisy correlations

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).