In ciphertext policy attribute-based encryption (CP-ABE), every secret key is associated with a set of attributes, and every ciphertext is associated with an access structure on attributes. Decryption is enabled if and only if the user's attribute set satisfies the ciphertext access structure. This provides fine-grained access control on shared data in many practical settings, e.g., secure database and IP multicast.

Task-PIOA is a modeling framework for distributed systems with both probabilistic and nondeterministic behaviors. It is suitable for cryptographic applications because its task-based scheduling mechanism is less powerful than the traditional perfect-information scheduler. Moreover, one can speak of two types of complexity restrictions: time bounds on description of task-PIOAs and time bounds on length of schedules. This distinction, along with the flexibility of nondeterministic specifications, are interesting departures from existing formal frameworks for computational security. The current paper presents a new approximate implementation relation for task-PIOAs. This relation is transitive and is preserved under hiding of external actions. Also, it is shown to be preserved under concurrent composition, with any polynomial number of substitutions. Building upon this foundation, we present the notion of structures, which classifies communications into two categories: those with a distinguisher environment and those with an adversary. We then formulate secure emulation in the spirit of traditional simulation-based security, and a composition theorem follows as a corollary of the composition theorem for the new approximate implementation relation.

We introduce a notion of finite testing, based on statistical hypothesis tests, via a variant of the well-known trace machine. Under this scenario, two processes are deemed observationally equivalent if they cannot be distinguished by any finite test. We consider processes modeled as image finite probabilistic automata and prove that our notion of observational equivalence coincides with the trace distribution equivalence proposed by Segala. Along the way, we give an explicit characterization of the set of probabilistic generalize the Approximation Induction Principle by defining an also prove limit and convex closure properties of trace distributions in an appropriate metric space.