We develop the dual system encryption methodology to provide fully secure functional encryption systems. We introduce new proof techniques and explore their applications, resulting in systems that advance the state of the art in terms of functionality, security, and efficiency. Our approach constructs versatile tools for adapting the dual system encryption methodology to new functionalities and efficiency goals. As particular demonstrations of our techniques, we obtain fully secure ciphertext-policy attribute-based encryption systems in the single authority and decentralized settings. Our work has provided the first fully secure attribute-based encryption schemes as well as the first decentralized schemes achieving desired levels of flexibility.