We put forward a new notion of pseudorandom functions (PRFs) we call
constrained PRFs. In a standard PRF there is a master key k that enables one to evaluate the function at all points in the domain of the
function. In a constrained PRF it is possible to derive constrained keys kS from the master key k. A constrained key kS enables the
evaluation of the PRF at a certain subset S of the domain and
nowhere else. We present a formal framework for this concept and show
that constrained PRFs can be used to construct powerful primitives such as identity-based key exchange and an optimal private broadcast
encryption system. We then construct constrained PRFs for several natural set systems needed for these applications. We conclude with several open problems relating to this new concept.