  Traditional approaches to Quantitative Information Flow (QIF) represent the
adversary's prior knowledge of possible secret values as a single probability
distribution. This representation may miss important structure. For instance,
representing prior knowledge about passwords of a system's users in this way
overlooks the fact that many users generate passwords using some strategy.
Knowledge of such strategies can help the adversary in guessing a secret, so
ignoring them may underestimate the secret's vulnerability. In this paper we
explicitly model strategies as distributions on secrets, and generalize the
representation of the adversary's prior knowledge from a distribution on
secrets to an environment, which is a distribution on strategies (and, thus, a
distribution on distributions on secrets, called a hyper-distribution). By
applying information-theoretic techniques to environments we derive several
meaningful generalizations of the traditional approach to QIF. In particular,
we disentangle the vulnerability of a secret from the vulnerability of the
strategies that generate secrets, and thereby distinguish security by
aggregation--which relies on the uncertainty over strategies--from security by
strategy--which relies on the intrinsic uncertainty within a strategy. We also
demonstrate that, in a precise way, no further generalization of prior
knowledge (e.g., by using distributions of even higher order) is needed to
soundly quantify the vulnerability of the secret.
