In a traditional public key encryption system, data is encrypted to a particular individual who is fixed by a public key, and the revealed information is all-or-nothing. Functional encryption (FE) gives a new way of sharing encrypted data, with fine-grained control over revealed information on encrypted data. In FE, a secret key enables a user to learn a specific function of the encrypted data and nothing else. Attribute-based encryption (ABE) can specify a wide variety of receiver sets by using relation between key and cipher text parameters. This survey for a wide audience will first give the notions of FE, ABE (and attribute-based signatures (ABS)), and describe several practical applications of FE schemes on dual pairing vector spaces (DPVS). The cryptosystems are implemented by using a pairing operation on an elliptic curve, which has achieved many new applications including FE in recent years. We also give a brief summary of the mathematical mechanism, e.g., The algorithm, parameter selection, and underlying security.