In this paper we investigate the topic of integrated public-key encryption (PKE) and public-key encryption with keyword search (PEKS) schemes (PKE-PEKS as shorthand). We first formalize the strongest security notion to date for PKE-PEKS schemes, named joint CCA-security. We then propose two simple constructions of jointly CCA-secure PKEPEKS schemes from anonymous (hierarchical) identity-based encryption schemes. Besides, we also define the notion of consistency for PKE-PEKS schemes, as well as revisit its related notions (including consistency of PEKS schemes, robustness and collision-freeness of IBE schemes), which may be of independent interest. ∗ State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China. Yu Chen and Dongdai Lin are supported by the National Natural Science Foundation of China under Grant No. 61303257, 61379141, the Strategic Priority Research Program of CAS under Grant No. XDA06010701,...