In this paper, the authors generalize the paradigm of Hash Proof System (HPS) proposed by Cramer and Shoup. In the central of their generalization, they lift subset membership problem to distribution distinguish problem. Their generalized HPS clarifies and encompass all the known Public-Key Encryption (PKE) schemes that essentially implement the idea of hash proof system. Moreover, besides existing smoothness property, they introduce an additional property named anonymity for HPS. As a natural application, they consider anonymity for PKE in the presence of key-leakage, and provide a generic construction of leakage-resilient anonymous PKE from anonymous HPS. They then extend their generalization to the identity-based setting.